Article pubs.acs.org/JPCB
How to Determine the Size of Folding Nuclei of Protofibrils from the Concentration Dependence of the Rate and Lag-Time of Aggregation. I. Modeling the Amyloid Protofibril Formation Nikita V. Dovidchenko,† Alexey V. Finkelstein,†,‡ and Oxana V. Galzitskaya*,† †
Institute of Protein Research, Russian Academy of Sciences, 4 Institutskaya str., Pushchino, Moscow Region, 142290, Russia Pushchino Branch of the Moscow State Lomonosov University, 4 Institutskaya str., Pushchino, Moscow Region, 142290, Russia
‡
S Supporting Information *
ABSTRACT: The question about the size of nuclei of formation of protofibrils (which constitute mature amyloid fibrils) formed by different proteins and peptides is yet open and debatable because of the absence of solid knowledge of underlying mechanisms of amyloid formation. In this work, a kinetic model of the process of formation of amyloid protofibrils is suggested, which allows calculation of the size of the nuclei using only kinetic data. In addition to the stage of primary nucleation, which is believed to be present in many protein aggregation processes, the given model includes both linear growth of protofibrils (proceeding only at the cost of attaching of monomers to the ends) and exponential growth of protofibrils at the cost of growth from the surface, branching, and fragmentation with the secondary nuclei. Theoretically, only the exponential growth is compatible with the existence of a pronounced lag-period (which can take much more time then the growth of aggregates themselves). The obtained analytical solution allows us to determine the size of the primary and secondary nuclei from the experimentally obtained concentration dependences of the time of growth and the new parameterthe ratio Lrel of the lag-time duration to the time of growth of amyloid protofibrils.
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INTRODUCTION Filamentous aggregates are quite ubiquitous phenomena involved in many aspects of living matterboth normal (actin, tubulin) and pathogenic (hemoglobin S aggregates in sickle cell anemia, amyloid fibrils). However, the understanding of the rate laws that underlies this phenomenon is a challenging task despite the wealth of experimental data. The first major advances in the understanding of the rate laws which drive protein polymerization reactions were done in the work by Oosawa et al.1 The authors have shown that the process of actin filament formation can be successfully approximated by a model which consists of two stages primary nucleation and consecutive joining of monomers (the model of the so-called “linear growth”). To the main results of this work, one should attribute the analytical solution of the rate equations, which has shown that, in such a kinetic scheme, the fibril mass accumulation during the initial aggregation phase (i.e., the time when still a small amount of aggregates exists) follows the quadratic dependence on time. Later this model was generalized as described in refs 2 and 3 for different special cases. However, the prediction of a quadratic dependence of fibril mass accumulation on time during the initial phase has failed for hemoglobin S in sickle cell anemia,4 which led to the next major step in the understanding of kinetic features on protein aggregation. To explain the observed effect of “extreme autocatalysis” and the strong concentration dependence, Ferrone et al. developed a model of “heterogeneous nucleation”.5 This model assumes the following events: first, © 2014 American Chemical Society
in the normal course of nucleation, a protofibril is formed, and then, on its surface, additional protofibrils can be formed. The concept of heterogeneous nucleation was a new and important contribution to the theory of protein aggregation. Moreover, it follows from this model that the amount of available surface for aggregation (addition of monomers) increases mainly due to the increase in the size of the aggregate (because the surface of protofibrils can also produce nuclei), which results in a comparatively long lag-period at the beginning of the process and a very rapid fibril mass accumulation (“exponential” growth, rather than “quadratic”) after the lag-period. Nowadays, the interest in the protein aggregation kinetics is mostly due to extensive studies of amyloid formation. Amyloids are protein aggregates, which usually have a fibrillar morphology, with an intermolecular cross-beta structure, and they can be stained by special dyes (Congo red, ThT).6 Amyloids are associated with diseases such as scrapie, mad-cow disease (however, the role of amyloid aggregates themselves in the diseases is not clear yet), Creutzfeldt−Jakob disease in humans, and many other infectious, sporadic, and hereditary diseases.7 Intricacy of mechanisms which drive the formation of amyloid aggregates should be addressed to the presence of signs of both linear and exponential growth scenarios such as long fibrillar structures (which can be consequences of Received: August 20, 2013 Revised: December 13, 2013 Published: January 9, 2014 1189
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period.14 For insulin, it has been shown that aggregation occurs by the lateral growth scenario, which is one of the types of branching.15 Branching in itself (when growing fibrils have a tree-like structure) was directly shown to be present in the process of amyloidogenesis of glucagon hormone peptide16 and Aβ-peptide.17,18 Recent experiments19 have shown that a βstructural droplet (as it would be in the case of growth from the surface) also might be one of the forms of amyloid aggregates; still, this requires a deeper analysis. In this article, we suggest a generalized model where different regimes of growth are considered. Same as Oosawa et al.,1 we assume that the growth of aggregates (for all scenarios considered in this paper) proceeds by adsorption of monomers floating in a rather diluted solution. Thus, we ignore reactions between the already adsorbed monomers. For simplicity, we assume also that a “monomer” includes only one protein molecule; however, there is an evident possibility to generalize our approach for the case when “monomers” floating in solution consist of several protein molecules. Despite the fact that such a model is simplified and various scenarios of growth of the aggregates were already analyzed in different papers,1,9,10,12 here we introduce some new important relations between experimentally measurable parameters (in particular, the relative duration of the lag-time, Lrel) and specific features of kinetic mechanisms of amyloid growth were derived. It will be proven that not all the scenarios of exponential growth (such as fragmentation, branching, and growth from the surface) can be distinguished from each other on the basis of kinetic experiments. This statement implies the requirement for additional, nonkinetic experimental data, which would directly point out what scenario of growth takes place in a particular case. However, the suggested herein model enables one to calculate the nucleus size (“nucleus” is the most unstable aggregate, which is a predecessor of amyloidal seedsthe smallest stable state aggregates) directly from the experimental data, which are important for determining the mechanism of the amyloid formation in each particular case.
consecutive monomer addiction to the ends of the growing fibril) paired to the rapid “exponential” growth of the aggregate. A sufficiently large number of mathematical models for polymerization reactions were proposed on the basis of a combination of the nucleation mechanism at the first stage of reaction and exponential aggregation at the second, where, often, a potential candidate for a probable mechanism of exponential growth is fragmentation.8,9 Radford and coauthors tried to approximate the kinetic behavior of β2-microglobulin aggregation with a “modular” system of kinetic equations.9 Thus, depending on the proposed mechanism, a set of blocks (“modules”) was selected. Examining the combination of various “modules”, a model with the best agreement with the experiment was chosen. This model included a module for the description of polymerization with sequenced monomer addition and a module for the fragmentation mechanism. Knowles and coauthors10 approximated equations which describe the process of amyloidogenesis analytically. Their model included the stage of nucleus formation and the stage of exponential growth with a fragmentation scenario. The authors noted that the process of fragmentation plays a key role in the approximation of experimental data. Also, the authors spotted that almost all the considered proteins (two of which, though, showed a bit different dependence), for which the kinetic data were approximated, showed linear scaling in the logarithmic coordinates (C/C0, Tlag/Tlag0, where C is the monomer concentration, Tlag is the duration of the lag period, and index 0 denotes the maximum value of the variable (C or Tlag) for the given set of kinetic curves) with the constant exponential coefficient (i.e., the dependence in ln Tlag ∼ const + γ ln[C], with a constant of γ = −1/2). Kohen et al.11 (based on ref 10) added the process of secondary nucleation (i.e., the ability to form a nucleus on the surface of growing fibrils) to the kinetic scheme with the fragmentation mechanism, and solved the resultant equations analytically. To explain the scaling effect observed in ref 10, the dependence of the secondary nucleus size from the lag-period was calculated. Thus, according to the authors’ model, γ is dependent on the secondary nucleus size n as γ = −(n + 1)/ 2. In the case of an experiment with a substantial lag-period and n = 0 (i.e., the absence of the secondary nucleation process), one has the solution for the fragmentation process, so that the scaling in ref 10 was addressed to the specific mechanism of amyloid formation reaction. Finke and Watski12 proposed a simple model to describe the process of amyloid aggregation where exponential growth was modeled by including the proportionality of the “ends” of the growing aggregate to the mass of the aggregate itself. Despite the fact that in this model the description of nonquadratic mass accumulation during early times of aggregate growth is indeed possible, the analytical solution represents a sigmoid curve, and in the experimental cases with a nonsigmoid kinetic curve, this model cannot be applied.13 Despite the variety of scenarios and models used to explain characteristic kinetic features of the amyloid self-assembly process, there are still questions to answer because of technical difficulties; it is not always clear how individual stages influence the course of the reaction in general, since one should use simplifications10 or numerically simulate the kinetic scheme with a very large number of parameters.9 Moreover, apparently, in nature one can meet all mechanisms of exponential growth. Thus, for prions (PrP), it is believed that fragmentation is the main factor of their spontaneous fast growth after a long lag-
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THEORY AND METHODS Glossary of Terms. The term “amyloid fibrils” refers to relatively straight protein fibrils that often (but not always) consist of multiple protofilaments twisted around the fibril axis. Protofilaments (or simply filaments): long straight filaments within the mature amyloid fibrils. Protofibrils: fibrillar, often curved-linear structures that appear in protein solutions before the mature amyloid fibrils appear. Consideration of the difference between “protofilaments” and “protofibrils” is beyond the scope of our kinetic study, which, actually, models the formation of separate protofibrils before their inclusion into the mature amyloid fibrils. To avoid a misunderstanding, we shall call “polymers” both of them. It should be noted, though, that these “polymers” become visible to some experimental methods only after their association in large fibrils; the latter process is assumed to be faster than the formation of “polymers”. Oligomer: macromolecular complexes of self-assembled polypeptides but not in the fibril/filament form. Transition Time and Size of the Lag-Period. We have found that it is convenient to express experimental kinetic results in terms of the relative value of the lag-period, Lrel, which is the ratio of the time of the lag-period Tlag time and the transition time T2 (see Figure 1). 1190
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P + M ⎯→ ⎯ P 2k 2
(more precisely, the sum of all Pn + M ⎯→ ⎯ Pn + 1 2k 2
with n ≥ nS). Here, k2 is the rate constant of this polymerization, while the reverse process Pn + M ← Pn+1 does not play a significant role in this irreversible process. These two stages are described by the following kinetic equations with respect to the monomer concentration [M] and the concentration of ends of all stable polymers [P] (the concentration of ends is twice greater than the concentration of polymers): ⎧ d[P] = 2k+ns[M]n * ⎪ ⎪ dt ⎨ ⎪ d[M] = −k+ns[M]n * − k 2[P][M] ⎪ ⎩ dt
Figure 1. Visual representation of the lag-period, transition time, and relative size Lrel of the lag-period.
The Model of Linear Growth. The simplest model, which describes the formation of amyloid protofibrils, describes a straight, successive, and irreversible (after the nucleus formation) reaction, where the growth is initiated by the ends of polymers (Figure 2). Such a model has been already
(1)
In the latter equation of this system (first obtained by Oosawa et al.1), one can neglect the small term −k+nS[M]n* (i.e., the monomers spent to form a seed), which is permissible after the very beginning of the process (at the times t ≫ t0 = nS/(k2[MΣ])), when the fraction of monomers involved in the seeds is much smaller than the fraction of monomers involved in the long nascent polymers. System 1 can be strictly solved for the whole process of the reaction (see the Supporting Information, item 1): 41/ n * exp( −t /T ) [M] = [MΣ] [1 + exp( −n*t /T )]2/ n *
where T = (n*/(4k2k+nS[MΣ]n*))1/2 is a characteristic time of the descent of [M] at t → ∞, i.e., at the end of the polymerization reaction. In the case of linear growth, we found Lrel to be dependent only on the size n* of a nucleus and independent of the protein concentration, while durations of various stages of the linear growth are proportional to the initial monomer concentration in the power of −n*/2 (see the Supporting Information, eqs S.1.6−S.1.10, and Table 1). The Lrel(n*) function is presented numerically (Figure 3). As can be seen from Figure 3, Lrel is a monotonically decreasing function limited from the above, so that the experimental values of Lrel > 0.2 cannot describe the model of linear growth. However, the experimental data revealed Lrel to be more than 0.2 in many cases,4,22−25 which means that the linear growth regime model is not applicable in such cases and one needs to introduce scenarios where the number of protofibrils increases exponentially. First Model of Exponential Growth: the “Growth from the Surface” Scenario. The model describing the formation of amyloid protofibrils as an irreversible reaction, initiated by any monomer unit involved in the aggregate state, is shown in Figure 4. It should be noted that, in this scenario of amyloidogenesis, the fibrils are not favorable structures by definition; rather, a droplet-like structure will be formed. However, because certain experiments show some signs of this scenario (nonfibrillar β-structure rich aggregates),19 it is of potential interest to try to model it. A similar scheme of reaction was mentioned previously in a work of Morris et al.;12 however, no detailed analysis was done.
Figure 2. Free energy change during the initiation and linear growth of a protofibril. Squares: free monomer. Balls: monomer in an aggregated form. “Nucleus” is the most unstable aggregate at the reaction pathway, and “seeds” is the smallest stable aggregate.
introduced by Oosawa et al.,1 and an analytical solution for appropriate system of equations was provided. However, for our analysis, it is convenient to rewrite equations in terms of characteristic times which were introduced in the previous section; thus, the key values would be obtained more naturally. In the approximation given by the transition state theory,20,21 a quasi-stationary irreversible process of polymerization is described by two kinetic phases: (1) The irreversible initiation of polymerization, i.e., formation of the seeds ns M → PnS k+
(the reverse process PnS → nSM plays no significant role if the concentration of seeds, which are rapidly involved in further polymerization, is low). The seed consists of nS monomers M; nS is the smallest number of monomers in the stable polymer P. The process goes with an effective rate constant k+, which is defined by the height of the free-energy barrier, corresponding to the nucleus comprising n* monomers; n* < nS (see Figure 2); each polymer P has two ends capable of binding new monomers M. (2) The further irreversible polymerization of monomers M at the ends of already formed protofibrils P 1191
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Table 1. Theoretical Dependencies of Parameters of Experimental Kinetic Curves on the Total Concentration of Monomers and the Size of the Nuclei scenario of growth linear growth
growth from the surface
dependence of the characteristic duration of the aggregate growth (T2) and of the relative lag-period duration (Lrel) on the total concentration of monomers [MΣ]
eqs S.1.7−S.1.9 (SupportingInformation)
n* = − 2
Lrel = const ’ < 0.2
eq S.1.10 (SupportingInformation), Figure 3 eq S.2.8, eqs S.2.9 and S.2.7 (SupportingInformation)
n* = 2 − 4
ln T2 = const − ln[MΣ]
{
Lrel = const′ −
bifurcation
direct formula for calculation of n* and n2 through the slopes of the experimental lines
n* ln[MΣ] 2 n* ln Tlag = const − ln[MΣ] + ln{const′} 2 ln T2 = const −
ln Tlag = const − ln[MΣ] + ln const′ −
fragmentation
references on equations and figures
d(ln T2) d(ln[MΣ])
n* − 2 ln[MΣ] 4
}
d(Lrel) d(ln[MΣ])
eq 3, eq S.2.11 (SupportingInformation)
(n* − 2) ln[MΣ] 4
1 ln[MΣ] 2 1 ln Tlag = const − ln[MΣ] + ln{const′ − (n* − 1) ln[MΣ]} 2
eq S.3.3, eqs S.3.8 and S.3.9 (SupportingInformation)
n* = 1 −
Lrel = const′ − (n* − 1) ln[MΣ]
eq 5, eq S.3.10 (SupportingInformation) eq S.4.3, eqs S.4.4 and S.3.9 (SupportingInformation)
n2 = −1 − 2
ln T2 = const −
n2 + 1 ln[MΣ] 2 n +1 ln Tlag = const − 2 ln[MΣ] + ln{const′ − (n* − n2 − 1) ln[MΣ]} 2
ln T2 = const −
d(Lrel) d(ln[MΣ])
d(ln T2) d(ln[MΣ])
n* = 1 + n2 −
d(Lrel) d(ln[MΣ])
eq S.4.5 (SupportingInformation)
Lrel = const′ − (n* − n2 − 1) ln[MΣ]
(1) The initiation of polymerization, i.e., formation of the seed ns M → PnS k+
As above, PnS is a minimal stable unit (consisting of nS monomers M), which forms with the effective rate constant k+, determined by the height of the free-energy barrier (i.e., a “nucleus” of n* monomers; n* < nS). (2) The sum of all aggregations of monomers at polymers Pn + M ⎯→ ⎯ Pn + 1 nk 2
when n ≥ nS. It is the further irreversible aggregation at the already formed aggregates in which the reverse process Pn + M ← Pn+1 does not play a significant role. The speed of the process is proportional to the number [P]all = [MΣ] − [M] of monomers in all aggregates. As a result, we obtain
Figure 3. Numerical calculation of Lrel as a function of the nucleus size n* (see eq S.1.10, Supporting Information).
d[M] = −2k+ns[M]n * − k 2([MΣ] − [M]) ·[M] dt
(2)
Equation 2 can be solved for the early stages of the reaction, where [MΣ] − [M] ≪ [MΣ], and thus [M] ≈ [MΣ] (see the Supporting Information, item 2):
Figure 4. Dependence of the free energy on the reaction coordinates in the process of amyloidogenesis where the growth occurs at the surface of aggregates.
(1 + λ)e−xt [M] = [MΣ] 1 + λ ·e−xt
Here λ = k2/(2k+ns[MΣ]n*−2), x = k2[MΣ](1 + 1/λ), and x ≡ 1/ T2, where x and T2 are the rate and time of the exponential growth, correspondingly.
In the approximation given by the transition state theory,20,21 a quasi-stationary irreversible aggregation process is described by two kinetic phases: 1192
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Lrel can be easily found when λ ≫ 1, which means that the rate of initiation of aggregation (connected with k+) is much smaller than the rate of growth of the aggregates (connected with k2): Lrel ≈
⎛ ⎞ ⎫ k2 1⎜ ⎧ ⎬ − 2⎟⎟ ln⎨ ⎜ n *−2 4 ⎝ ⎩ 2k+ns[MΣ] ⎭ ⎠ ⎪
⎪
⎪
⎪
A protofibril divides into two stable parts; the chance of fragmentation for a single protofibril is roughly proportional to its length, lP, with the rate constant λ+. The possibility to merge two protofibrils into one (with the rate constant λ−) is not important at a low concentration of protofibrils at the beginning of the process. These stages are described by the following equations with respect to the concentration [M] of monomers M and the concentration [P] of ends of all the stable polymers P:
(3)
(see the Supporting Information, item 2). The important conclusion that follows is that there is no upper limit of Lrel (as it was in the case of the linear regime of growth, see Figure 3) when [MΣ] → 0 and n* > 2 or [MΣ] → ∞ and n* < 2, while at n* = 2 Lrel does not depend on [MΣ] (see Table 1). A similar conclusion (as it would be shown further) is applicable in the case of other forms of exponential regime of growth of amyloid protofibrils. Another Model of Exponential Growth: the “Fragmentation” Scenario. This model also describing the formation of amyloid as irreversible polymerization is shown in Figure 5. This scheme was first used for a description of the
⎧ d[P] = 2k+[M]n * + 2λ+([MΣ] − [M]) ⎪ d t ⎪ − λ−[P]2 ⎨ ⎪ d[M] ⎪ = −k+ns[M]n * − k 2[P][M] ⎪ ⎩ dt
(4) 5,27
In the latter equation of this system (cf. Ferrone et al. and Wegner26), one can neglect (as in system 1) the relatively small term −k+ns[M]n*; also, at the beginning of the process, [MΣ] − [M] ≪ [MΣ], and [P] is small; thus, λ−[P]2 can be neglected in the first equation, while [M] can be replaced by [MΣ] in k+[M]n* and in k2[P][M]. Then, ⎧ d[P] ≈ 2k+[MΣ]n * + 2λ+([MΣ] − [M]) ⎪ ⎪ dt ⎨ ⎪ d[M] ≈ −k 2[P][MΣ] ⎪ ⎩ dt
(4a)
for early stages of the reaction; solution of this system (see Supporting Information, item 3) gives
Figure 5. Dependence of the free energy of the reaction coordinates in the process of amyloidogenesis with fragmentation.
[M] = 1 − A(et / T2 + e−t / T2 − 2) [MΣ]
kinetics of actin filament formation;26 however, analytical solution was obtained only in a paper by Knowles et al. in 2009.10 Because of consideration of secondary nucleation and fragmentation cases together, the authors’ analysis lacked major relations, which could be used for determination of nucleus size, n*, so in this paper we will reconsider both cases separately. In the approximation given by the transition state theory,20,21 a quasi-stationary process of irreversible polymerization with fragmentation is described by three kinetic phases: (1) The initiation of polymerization (the same as in the reactions shown in Figures 3 and 4) ns M → PnS
where A = k+[MΣ]n*−1/2λ+ and T2 = 1/(4λ+k2[MΣ])1/2. We found L rel from this solution (see Supporting Information, item 3): ⎛ ⎞ 1 λ+ ⎟− Lrel = ln⎜⎜ n *−1 ⎟ 2 ⎝ k+[MΣ] ⎠
(5)
It is clearly seen that the dependence of Lrel on [MΣ] vanishes when n* = 1 (see Table 1). Moreover, in contrast to the model of the “growth from the surface scenario”, where Lrel decreases with an increase of [MΣ] when n* > 2, in the “fragmentation scenario”, Lrel decreases with an increase of [MΣ] when n* > 1, and a case where Lrel increases with an increase of [MΣ] is absent (because n* cannot be less than 1). One More Model of Exponential Growth: the “Bifurcation” Scenario. A model describing the formation of amyloid as a branching reaction, where the branching is initiated by the secondary nucleus attached to the inner parts of the polymer and the following growth occurs at the ends of the polymer and its branches, is shown in Figure 6. Initially, a similar model applied to biological matter was the description of hemoglobin aggregation during sickle cell anemia decease.4 However, an analytical solution was provided (as in a case of fragmentation scenario) in the paper by Knowles et al.;10 still, as it was mentioned above, the provided analysis lacked completeness, so in this paper we will reconsider the scheme. The process of irreversible polymerization with chain branching is described by three kinetic phases, only one of
k+
(2) The further irreversible (as in Figures 3 and 4) reaction of polymerization on the ends of the already formed protofibrils, where the reverse process Pn + M ← Pn+1 does not play a significant role P + M ⎯→ ⎯ P 2k 2
(or, more precisely, the sum of Pn + M ⎯→ ⎯ Pn + 1 2k 2
for all reactions with n ≥ nS). (3) A fragmentation of protofibrils λ−
P ←→ P + P λ+lP
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⎛ ⎞ 1 λ+′ ⎟− Lrel = ln⎜⎜ n *−n2 − 1 ⎟ 2 ⎝ 2k+[MΣ] ⎠
(8)
The dependence of Lrel on [MΣ] vanishes when n* − n2 = 1, and Lrel increases with an increase of [MΣ] when n* − n2 < 1 and decreases when n* − n2 > 1. It should be noted that the case of n2 = 0 is quite possible: this case means that the initiation energy is connected with deformation of the protofibril rather than with the deformation of the adhering monomer. This case will be experimentally investigated in our following work. Analysis of Concentration Dependencies. As it was mentioned above, the considered schemes of aggregation kinetics result in formulas by which one is able to calculate the size of the protofibril nucleus, n*, and (in the case of a bifurcation scenario) of the branched nucleus, n2. Such a calculation can be made through the analysis of the concentration dependences of two kinetic characteristics: the transition time (T2) and the ratio of duration of the lag-period to the transition time. These characteristics have a simple connection with the initial concentrations of monomers (see Table 1), and therefore, they are more appropriate for obtaining the sizes of primary and secondary nuclei than the often used (see, e.g., ref 10) duration of the lag-period itself. For these calculations, it is necessary to draw an array of points which reflects Lrel and ln(T2) for given molecules as a function of the initial monomer concentration [MΣ] (see Table 1 and the section “Transition Time and Size of the LagPeriod”). To calculate n* or n2, the obtained array should be approximated by a straight line using the method of leastsquares (see Figure 7). The error of the tangent coefficient is estimated as ε = tα0,(n−2)[(1/(n − 2))∑ni=1(yi − ŷi)2)/(∑ni=1(xi − x)̅ 2)]1/2,28 where n is the number of experimental [MΣ] values,
Figure 6. Dependence of the free energy on the reaction coordinates in the process of amyloidogenesis with bifurcations.
which is only slightly different from polymerization with fragmentation; compare Figures 5 and 6: (1) The initiation of polymerization (the same as in Figures 3−5) ns M → PnS k+
(2) The further irreversible (as in Figures 3−5) polymerization on the ends of already formed protofibrils and their branches Pn + M ⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 ends × k 2
(more precisely, the sum of all Pn + M ⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 ends × k 2
with n ≥ nS). (3) The emergence of a stable second protofibril on the first one (the speed of the emergence of such a protofibril is proportional to the length of the base protofibril, lP, and in total over all protofibrilsto the number of monomers in them)
P ⎯→ ⎯ P, P′ λ′+lP
Unlike the fragmentation process shown in Figure 5, which adds to the system two points of growth (two ends of the protofibril), the branching adds only one point of growth (end of the protofibril P′). The beginning of the branching process, when [MΣ] − [M] ≪ [MΣ] and [P], the concentration of ends of all the stable polymers, is also very small, is described by equations that are almost identical to those describing the beginning of the fragmentation (see eq 4a): ⎧ d[P] ≈ 2k+[MΣ]n * + λ+′ [MΣ]n2 ([MΣ] − [M]) ⎪ ⎪ dt ⎨ ⎪ d[M] ≈ −k 2[P][MΣ] ⎪ ⎩ dt
(6)
with the substitution 2λ+ ⇒ λ+′ [MΣ] , where λ+′ is no longer the rate constant of fragmentation but rather the bifurcation rate constant and n2 is the effective number of monomers in the nucleus of a branch. Therefore, the whole mathematical description of the beginning of the process of branching coincides with the description of the fragmentation process (see Supporting Information, item 3, and Table 1): n2
[M] = 1 − A(et / T2 + e−t / T2 − 2) [MΣ]
where A = k+[MΣ] while
n*−n2−1
/λ+′ and T2 = 1/(2λ+′ k2[MΣ]
Figure 7. ln T2 and Lrel versus ln[MΣ] for Yeast Prion Sup3523 with the approximation lines calculated by the least-squares method. The tangent coefficient kT2 = d(ln T2)/d(ln[MΣ]) of the approximation line is kT2 = −0.24 ± 0.45, and the tangent coefficient kLrel = d(Lrel)/ d(ln[MΣ]) of the approximation line is kLrel = −0.09 ± 0.20.
(7) n2+1
])1/2, 1194
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Table 2. The Range of Lrel and ln T2 Values and the Nuclei Sizes n*, n2 Calculated from the Experimental Data protein or peptide and a reference to the experimental study of the amyloid formation by this protein or peptide insulin22 17
Aβ42 Yeast Prion Sup3523 Yeast Prion Ure2p24 murine FBP28 WW domain25 β2-microglobulin29 27th immunoglobulin domain from human cardiac titin (TI I27)29 apolipoprotein C-II30
model that follows from the Lrel and T2 calculation
Lrel estimated at various ln[MΣ] (min − max)a
ln T2 estimated at various ln[MΣ] (min − max)
n* ± εn*, primary nucleusb
n2 ± εn2, secondary nucleusb
exponential (fragmentation/ bifurcation) bifurcation exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) bifurcation
5.17−5.56
−0.49−0.08
1.13 ± 0.19
−0.48 ± 0.16
0.53−0.67 0.29−0.70
−0.51−1.44 4.62−5.52
2.64 ± 0.11 1.09 ± 0.20
1.72 ± 0.05 −0.51 ± 0.45
0.76−1.02
1.79−2.39
0.96 ± 1.52
−0.23 ± 1.40
1.56−2.10
4.76−6.02
1.21 ± 1.27
0.05 ± 1.02
1.48−3.86
0.79−2.28
1.58 ± 0.58
−0.06 ± 0.18
0.14−0.34
5.93−7.85
2.86 ± 0.30
2.04 ± 0.29
0.06−0.10
2.77−4.82
linear
a
The errors are computed using the Student’s coefficient corresponding to 0.95 confidence level. amyloidogenesis kinetics were kindly provided by S. E. Radford.
yi is the value of Lrel (or ln T2) corresponding to the ith value of [MΣ], ŷi is the approximated value of yi, xi̅ ≡ [MΣ]i, x̅ is the averaged [MΣ]i value, and tα0,(n−2) is the Student’s coefficient with confidence α0 = 0.95 and n − 2 degrees of freedom.
b
4.44 ± 0.38
Numerical data for β2-microglobulin
In ref 10, it was noted that most of the experimental data in the appropriate coordinates (with our definitions - (ln Tlag, ln[MΣ])) lies close to the line ln Tlag ∼ −(1/2) ln[MΣ] + const. Moreover, almost all studied proteins had Lrel ≫ 0.2, which means that their aggregates had an exponential mechanism of growth. This intriguing feature compelled us to take the existing experimental data on amyloid formation kinetics (see Table 1) and analyze them with our model. Table 1 suggested that a linear dependence ln Tlag ∼ −(1/2) ln[MΣ] + const can correspond only to the cases of the fragmentation or bifurcation scenarios with n* = 1 and n2 = 0. The same conclusion is obtained from the analysis of Lrel values as a function of MΣ. For the proteins insulin,22 Sup35,23 Ure2p,24 WW,25 Aβpeptide,17 and β2-microglobulin,29 it is definitely an exponential scenario of protofibril growth (Lrel ≫ 0.2). Moreover, the determination of n* directly from the experimental data in proposed coordinates (Lrel, ln[MΣ]) gives n* ≈ 1 (within the error) for all of these proteins (except for Aβ-peptide) with an exponential mechanism of growth (see Table 2). Aβ-peptide was the only object among those mentioned above which directly points toward the bifurcation scenario of growth having the primary nucleus size equal to n* ≈ 3 and the secondary nucleus size equal to n2 ≈ 2. In all the other cases, the secondary nucleus size is close to 0 (within errors), with a possible exception of insulin, where the small negative value of n2 may mean that some protofibrils lose monomers in the process of fragmentation or bifurcation. Thus, our results are in reasonable accordance with those of the authors whose data we used.17 Among all proteins used in ref 10 (and listed in Table 2), TI I2729 protein and apolipoprotein C-II30 got very low Lrel (0.25 ± 0.12 and 0.07 ± 0.03), which is typical of the linear growth scenario. Calculation of the nucleus size for apolipoprotein CII30 gives n* ≈ 4. However, the concentration dependence of Lrel on [M] for TI cannot be explained in terms of the linear growth scenario, so we applied an exponential mechanism analysis for this protein. Calculation showed the bifurcation scenario to be an adequate candidate for approximation of the experimental data. The corresponding dependences (ln T2, ln[MΣ]) and (Lrel, ln[MΣ]) give the primary nucleus size n* ≈ 2 and secondary nucleus size n2 ≈ 3, respectively.
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RESULTS AND DISCUSSION All the obtained dependencies are summarized in Table 1, and the procedure of determining what mechanism of amyloid formation is present in a given experiment should be done in the following way. If Lrel > 0.2 independently on [MΣ] the model of amyloid protofibril formation through the linear regime of growth is not applicable, and it is a clear case of the exponential mechanism of growth. Further determination of what exact scenario of exponential mechanism of growth is present in a given experiment can be done on the basis of the dependence of ln T2 from ln[MΣ] (see Table 1). The dependence of ln T2 on ln[MΣ] is different for growth from the surface scenario, for fragmentation, and for bifurcation scenarios. However, it is impossible to distinguish the bifurcation from fragmentation scenarios from kinetics, because all the kinetic dependencies for them have identical forms, if the size of the secondary nucleus is equal to zero. This statement implies the need for direct experimental data to determine if it is a fragmentation or bifurcation scenario. As it can be seen from Table 1, all the predicted dependencies of Lrel and ln T2 on ln[MΣ] are linear (unlike the ln Tlag on ln[MΣ] dependence), which provides a convenient possibility for calculation of the sizes n* and n2 of primary and secondary nuclei from the kinetic experimental data presented in coordinates (ln T2, ln[MΣ]) and (Lrel, ln[MΣ]) using the least-squares method. It should be reiterated that, for simplicity, we assumed that a “monomer” floating in solution includes only one protein molecule; if the floating “monomers” consist of nmono protein molecules (which can be established only experimentally), the values n* and n2 (which give the size of the nuclei in the number of “monomers”) must be multiplied by nmono to obtain the size of the nuclei in the numbers of protein molecules. In the case of exponential growth, it is better to use the Lrel dependence on ln[MΣ] and the ln T2 dependence on ln[MΣ] instead of ln Tlag on ln[MΣ] (used in ref 10), because in the first case the dependence has a linear form for every scenario of exponential growth. 1195
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(7) Cali, I.; Castellani, R.; Yuan, J.; Al-Shekhlee, A.; Cohen, M. L.; Xiao, X.; Moleres, F. J.; Parchi, P.; Zou, W. Q.; Gambetti, P. Classification of Sporadic Creutzfeldt-Jakob Disease Revisited. Brain 2006, 129, 2266−2277. (8) Serio, T. R.; Cashikar, A. G.; Kowal, A. S.; Sawicki, G. J.; Moslehi, J. J.; Serpell, L.; Arnsdorf, M. F.; Lindquist, S. L. Nucleated Conformational Conversion and the Replication of Conformational Information by a Prion Determinant. Science 2000, 289, 1317−1321. (9) Xue, W. F.; Homans, S. W.; Radford, S. E. Systematic Analysis of Nucleation-Dependent Polymerization Reveals New Insights into the Mechanism of Amyloid Self-Assembly. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 8926−8931. (10) Knowles, T. P.; Waudby, C. A.; Devlin, G. L.; Cohen, S. I.; Aguzzi, A.; Vendruscolo, M.; Terentjev, E. M.; Welland, M. E.; Dobson, C. M. An Analytical Solution to the Kinetics of Breakable Filament Assembly. Science 2009, 326, 1533−1537. (11) Cohen, S. I.; Vendruscolo, M.; Welland, M. E.; Dobson, C. M.; Terentjev, E. M.; Knowles, T. P. Nucleated Polymerization with Secondary Pathways. I. Time Evolution of the Principal Moments. J. Chem. Phys. 2011, 135, 065105. (12) Morris, A. M.; Watzky, M. A.; Finke, R. G. Protein Aggregation Kinetics, Mechanism, and Curve-Fitting: A Review of the Literature. Biochim. Biophys. Acta 2009, 1794, 375−397. (13) Giehm, L.; Otzen, D. E. Strategies to Increase the Reproducibility of Protein Fibrillization in Plate Reader Assays. Anal. Biochem. 2010, 400, 270−281. (14) Masel, J.; Jansen, V. A.; Nowak, M. A. Quantifying the Kinetic Parameters of Prion Replication. Biophys. Chem. 1999, 77, 139−152. (15) Loksztejn, A.; Dzwolak, W. Vortex-Induced Formation of Insulin Amyloid Superstructures Probed by Time-Lapse Atomic Force Microscopy and Circular Dichroism Spectroscopy. J. Mol. Biol. 2010, 395, 643−655. (16) Andersen, C. B.; Yagi, H.; Manno, M.; Martorana, V.; Ban, T.; Christiansen, G.; Otzen, D. E.; Goto, Y.; Rischel, C. Branching in Amyloid Fibril Growth. Biophys. J. 2009, 96, 1529−1536. (17) Cohen, S. I.; Linse, S.; Luheshi, L. M.; Hellstrand, E.; White, D. A.; Rajah, L.; Otzen, D. E.; Vendruscolo, M.; Dobson, C. M.; Knowles, T. P. Proliferation of Amyloid-β42 Aggregates Occurs through a Secondary Nucleation Mechanism. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 9758−9763. (18) Jeong, J. S.; Ansaloni, A.; Mezzenga, R.; Lashuel, H. A.; Dietler, G. Novel Mechanistic Insight into the Molecular Basis of Amyloid Polymorphism and Secondary Nucleation During Amyloid Formation. J. Mol. Biol. 2013, 425, 1765−1781. (19) Cho, K. R.; Huang, Y.; Yu, S.; Yin, S.; Plomp, M.; Qiu, S. R.; Lakshminarayanan, R.; Moradian-Oldak, J.; Sy, M.-S.; De Yoreo, J. J. A Multistage Pathway for Human Prion Protein Aggregation in Vitro: From Multimeric Seeds to β Oligomers and Nonfibrillar Structures. J. Am. Chem. Soc. 2011, 133, 8586−8593. (20) Eyring, H. The Activated Complex in Chemical Reactions. J. Chem. Phys. 1935, 3, 107−115. (21) Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; Wiley: Chichester, England 1981. (22) Fodera, V.; Librizzi, F.; Groenning, M.; van de Weert, M.; Leone, M. Secondary Nucleation and Accessible Surface in Insulin Amyloid Fibril Formation. J. Phys. Chem. B 2008, 112, 3853−3858. (23) Collins, S. R.; Douglass, A.; Vale, R. D.; Weissman, J. S. Mechanism of Prion Propagation: Amyloid Growth Occurs by Monomer Addition. PLoS Biol. 2004, 2, 1582−1590. (24) Zhu, L.; Zhang, X. J.; Wang, L. Y.; Zhou, J. M.; Perrett, S. Relationship between Stability of Folding Intermediates and Amyloid Formation for the Yeast Prion Ure2p: A Quantitative Analysis of the Effects of pH and Buffer System. J. Mol. Biol. 2003, 328, 235−254. (25) Ferguson, N.; Berriman, J.; Petrovich, M.; Sharpe, T. D.; Finch, J. T.; Fersht, A. R. Rapid Amyloid Fiber Formation from the FastFolding WW Domain Fbp28. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 9814−9819. (26) Wegner, A. Spontaneous Fragmentation of Actin Filaments in Physiological Conditions. Nature 1982, 296, 266−267.
CONCLUSIONS We have shown that kinetic curves can give insight into various mechanisms of amyloid aggregate formation. The first indicator of a possible regime of growth can be derived from Lrel, where the concentration-independent Lrel ≈ 0.2 is the border between linear and exponential scenarios of protofibril growth. Some conclusion can also be made about a kind of the exponential growth scenario which presents the given experiment (e.g., growth from the surface or fragmentation/bifurcation scenario; the distinction of the two latter scenarios requires direct approaches (for example, real-time observations of growing protofibrils at a single-protofibril level, as in ref 16)). The suggested theory of concentration-dependent kinetics can also reveal such important parameters of the reaction as the sizes of the primary and secondary folding nuclei of protofibrils. In the very end, we should add that the above given models correspond to “pure” cases of the amyloid formation via different mechanisms, while a mixture of several mechanisms can be observed sometimes in experiments.18
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ASSOCIATED CONTENT
S Supporting Information *
Solution of systems of kinetic equations for all discussed aggregation scenarios. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Phone/fax: +74967-318275/ +74967-318435. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are grateful to Prof. S. E. Radford who kindly provided us with numerical data for β2-microglobulin amyloidogenesis kinetics. We thank T. B. Kuvshinkina for assistance in the manuscript preparation. This study was supported in part by the Russian Foundation for Basic Research (Grant Nos. 11-0400763a and 13-04-00253a), Russian Academy of Sciences (programs “Molecular and Cell Biology” (Grant Nos. 01201353567 and 01201358029) and “Fundamental Sciences to Medicine”).
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REFERENCES
(1) Oosawa, F.; Asakura, S.; Hotta, K.; Imai, N.; Ooi, T. G-F Transformation of Actin as a Fibrous Condensation. J. Polym. Sci. 1959, 37, 323−336. (2) Frieden, C.; Goddette, D. W. Polymerization of Actin and ActinLike Systems: Evaluation of the Time Course of Polymerization in Relation to the Mechanism. Biochemistry 1983, 22, 5836−5843. (3) Goldstein, R. F.; Stryer, L. Cooperative Polymerization Reactions. Analytical Approximations, Numerical Examples, and Experimental Strategy. Biophys. J. 1986, 50, 583−599. (4) Hofrichter, J.; Ross, P. D.; Eaton, W. A. Kinetics and Mechanism of Deoxyhemoglobin S Gelation: A New Approach to Understanding Sickle Cell Disease. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 4864−4868. (5) Ferrone, F. A.; Hofrichter, J.; Sunshine, H. R.; Eaton, W. A. Kinetic Studies on Photolysis-Induced Gelation of Sickle Cell Hemoglobin Suggest a New Mechanism. Biophys. J. 1980, 32, 361− 380. (6) Buxbaum, J. N.; Linke, R. P. A Molecular History of the Amyloidoses. J. Mol. Biol. 2012, 421, 142−159. 1196
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(27) Bishop, M. F.; Ferrone, F. A. Kinetics of Nucleation-Controlled Polymerization. A Perturbation Treatment for Use with a Secondary Pathway. Biophys. J. 1984, 46, 631−644. (28) Kenney, J. F.; Keeping, E. S. Linear Regression and Correlation; Van Nostrand: Princeton, NJ, 1962. (29) Wright, C. F.; Teichmann, S. A.; Clarke, J.; Dobson, C. M. The Importance of Sequence Diversity in the Aggregation and Evolution of Proteins. Nature 2005, 438, 878−881. (30) Binger, K. J.; Pham, C. L. L.; Wilson, L. M.; Bailey, M. F.; Lawrence, L. J.; Schuck, P.; Howlett, G. J. Apolipoprotein C-II Amyloid Fibrils Assemble Via a Reversible Pathway That Includes Fibril Breaking and Rejoining. J. Mol. Biol. 2008, 376, 1116−1129.
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How to Determine the Size of Folding Nuclei of Protofibrils from the Concentration Dependence of the Rate and Lag-Time of Aggregation. I. Modeling the Amyloid Protofibril Formation Nikita V. Dovidchenko,† Alexey V. Finkelstein,†,‡ and Oxana V. Galzitskaya*,† †
Institute of Protein Research, Russian Academy of Sciences, 4 Institutskaya str., Pushchino, Moscow Region, 142290, Russia Pushchino Branch of the Moscow State Lomonosov University, 4 Institutskaya str., Pushchino, Moscow Region, 142290, Russia
‡
S Supporting Information *
ABSTRACT: The question about the size of nuclei of formation of protofibrils (which constitute mature amyloid fibrils) formed by different proteins and peptides is yet open and debatable because of the absence of solid knowledge of underlying mechanisms of amyloid formation. In this work, a kinetic model of the process of formation of amyloid protofibrils is suggested, which allows calculation of the size of the nuclei using only kinetic data. In addition to the stage of primary nucleation, which is believed to be present in many protein aggregation processes, the given model includes both linear growth of protofibrils (proceeding only at the cost of attaching of monomers to the ends) and exponential growth of protofibrils at the cost of growth from the surface, branching, and fragmentation with the secondary nuclei. Theoretically, only the exponential growth is compatible with the existence of a pronounced lag-period (which can take much more time then the growth of aggregates themselves). The obtained analytical solution allows us to determine the size of the primary and secondary nuclei from the experimentally obtained concentration dependences of the time of growth and the new parameterthe ratio Lrel of the lag-time duration to the time of growth of amyloid protofibrils.
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INTRODUCTION Filamentous aggregates are quite ubiquitous phenomena involved in many aspects of living matterboth normal (actin, tubulin) and pathogenic (hemoglobin S aggregates in sickle cell anemia, amyloid fibrils). However, the understanding of the rate laws that underlies this phenomenon is a challenging task despite the wealth of experimental data. The first major advances in the understanding of the rate laws which drive protein polymerization reactions were done in the work by Oosawa et al.1 The authors have shown that the process of actin filament formation can be successfully approximated by a model which consists of two stages primary nucleation and consecutive joining of monomers (the model of the so-called “linear growth”). To the main results of this work, one should attribute the analytical solution of the rate equations, which has shown that, in such a kinetic scheme, the fibril mass accumulation during the initial aggregation phase (i.e., the time when still a small amount of aggregates exists) follows the quadratic dependence on time. Later this model was generalized as described in refs 2 and 3 for different special cases. However, the prediction of a quadratic dependence of fibril mass accumulation on time during the initial phase has failed for hemoglobin S in sickle cell anemia,4 which led to the next major step in the understanding of kinetic features on protein aggregation. To explain the observed effect of “extreme autocatalysis” and the strong concentration dependence, Ferrone et al. developed a model of “heterogeneous nucleation”.5 This model assumes the following events: first, © 2014 American Chemical Society
in the normal course of nucleation, a protofibril is formed, and then, on its surface, additional protofibrils can be formed. The concept of heterogeneous nucleation was a new and important contribution to the theory of protein aggregation. Moreover, it follows from this model that the amount of available surface for aggregation (addition of monomers) increases mainly due to the increase in the size of the aggregate (because the surface of protofibrils can also produce nuclei), which results in a comparatively long lag-period at the beginning of the process and a very rapid fibril mass accumulation (“exponential” growth, rather than “quadratic”) after the lag-period. Nowadays, the interest in the protein aggregation kinetics is mostly due to extensive studies of amyloid formation. Amyloids are protein aggregates, which usually have a fibrillar morphology, with an intermolecular cross-beta structure, and they can be stained by special dyes (Congo red, ThT).6 Amyloids are associated with diseases such as scrapie, mad-cow disease (however, the role of amyloid aggregates themselves in the diseases is not clear yet), Creutzfeldt−Jakob disease in humans, and many other infectious, sporadic, and hereditary diseases.7 Intricacy of mechanisms which drive the formation of amyloid aggregates should be addressed to the presence of signs of both linear and exponential growth scenarios such as long fibrillar structures (which can be consequences of Received: August 20, 2013 Revised: December 13, 2013 Published: January 9, 2014 1189
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period.14 For insulin, it has been shown that aggregation occurs by the lateral growth scenario, which is one of the types of branching.15 Branching in itself (when growing fibrils have a tree-like structure) was directly shown to be present in the process of amyloidogenesis of glucagon hormone peptide16 and Aβ-peptide.17,18 Recent experiments19 have shown that a βstructural droplet (as it would be in the case of growth from the surface) also might be one of the forms of amyloid aggregates; still, this requires a deeper analysis. In this article, we suggest a generalized model where different regimes of growth are considered. Same as Oosawa et al.,1 we assume that the growth of aggregates (for all scenarios considered in this paper) proceeds by adsorption of monomers floating in a rather diluted solution. Thus, we ignore reactions between the already adsorbed monomers. For simplicity, we assume also that a “monomer” includes only one protein molecule; however, there is an evident possibility to generalize our approach for the case when “monomers” floating in solution consist of several protein molecules. Despite the fact that such a model is simplified and various scenarios of growth of the aggregates were already analyzed in different papers,1,9,10,12 here we introduce some new important relations between experimentally measurable parameters (in particular, the relative duration of the lag-time, Lrel) and specific features of kinetic mechanisms of amyloid growth were derived. It will be proven that not all the scenarios of exponential growth (such as fragmentation, branching, and growth from the surface) can be distinguished from each other on the basis of kinetic experiments. This statement implies the requirement for additional, nonkinetic experimental data, which would directly point out what scenario of growth takes place in a particular case. However, the suggested herein model enables one to calculate the nucleus size (“nucleus” is the most unstable aggregate, which is a predecessor of amyloidal seedsthe smallest stable state aggregates) directly from the experimental data, which are important for determining the mechanism of the amyloid formation in each particular case.
consecutive monomer addiction to the ends of the growing fibril) paired to the rapid “exponential” growth of the aggregate. A sufficiently large number of mathematical models for polymerization reactions were proposed on the basis of a combination of the nucleation mechanism at the first stage of reaction and exponential aggregation at the second, where, often, a potential candidate for a probable mechanism of exponential growth is fragmentation.8,9 Radford and coauthors tried to approximate the kinetic behavior of β2-microglobulin aggregation with a “modular” system of kinetic equations.9 Thus, depending on the proposed mechanism, a set of blocks (“modules”) was selected. Examining the combination of various “modules”, a model with the best agreement with the experiment was chosen. This model included a module for the description of polymerization with sequenced monomer addition and a module for the fragmentation mechanism. Knowles and coauthors10 approximated equations which describe the process of amyloidogenesis analytically. Their model included the stage of nucleus formation and the stage of exponential growth with a fragmentation scenario. The authors noted that the process of fragmentation plays a key role in the approximation of experimental data. Also, the authors spotted that almost all the considered proteins (two of which, though, showed a bit different dependence), for which the kinetic data were approximated, showed linear scaling in the logarithmic coordinates (C/C0, Tlag/Tlag0, where C is the monomer concentration, Tlag is the duration of the lag period, and index 0 denotes the maximum value of the variable (C or Tlag) for the given set of kinetic curves) with the constant exponential coefficient (i.e., the dependence in ln Tlag ∼ const + γ ln[C], with a constant of γ = −1/2). Kohen et al.11 (based on ref 10) added the process of secondary nucleation (i.e., the ability to form a nucleus on the surface of growing fibrils) to the kinetic scheme with the fragmentation mechanism, and solved the resultant equations analytically. To explain the scaling effect observed in ref 10, the dependence of the secondary nucleus size from the lag-period was calculated. Thus, according to the authors’ model, γ is dependent on the secondary nucleus size n as γ = −(n + 1)/ 2. In the case of an experiment with a substantial lag-period and n = 0 (i.e., the absence of the secondary nucleation process), one has the solution for the fragmentation process, so that the scaling in ref 10 was addressed to the specific mechanism of amyloid formation reaction. Finke and Watski12 proposed a simple model to describe the process of amyloid aggregation where exponential growth was modeled by including the proportionality of the “ends” of the growing aggregate to the mass of the aggregate itself. Despite the fact that in this model the description of nonquadratic mass accumulation during early times of aggregate growth is indeed possible, the analytical solution represents a sigmoid curve, and in the experimental cases with a nonsigmoid kinetic curve, this model cannot be applied.13 Despite the variety of scenarios and models used to explain characteristic kinetic features of the amyloid self-assembly process, there are still questions to answer because of technical difficulties; it is not always clear how individual stages influence the course of the reaction in general, since one should use simplifications10 or numerically simulate the kinetic scheme with a very large number of parameters.9 Moreover, apparently, in nature one can meet all mechanisms of exponential growth. Thus, for prions (PrP), it is believed that fragmentation is the main factor of their spontaneous fast growth after a long lag-
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THEORY AND METHODS Glossary of Terms. The term “amyloid fibrils” refers to relatively straight protein fibrils that often (but not always) consist of multiple protofilaments twisted around the fibril axis. Protofilaments (or simply filaments): long straight filaments within the mature amyloid fibrils. Protofibrils: fibrillar, often curved-linear structures that appear in protein solutions before the mature amyloid fibrils appear. Consideration of the difference between “protofilaments” and “protofibrils” is beyond the scope of our kinetic study, which, actually, models the formation of separate protofibrils before their inclusion into the mature amyloid fibrils. To avoid a misunderstanding, we shall call “polymers” both of them. It should be noted, though, that these “polymers” become visible to some experimental methods only after their association in large fibrils; the latter process is assumed to be faster than the formation of “polymers”. Oligomer: macromolecular complexes of self-assembled polypeptides but not in the fibril/filament form. Transition Time and Size of the Lag-Period. We have found that it is convenient to express experimental kinetic results in terms of the relative value of the lag-period, Lrel, which is the ratio of the time of the lag-period Tlag time and the transition time T2 (see Figure 1). 1190
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P + M ⎯→ ⎯ P 2k 2
(more precisely, the sum of all Pn + M ⎯→ ⎯ Pn + 1 2k 2
with n ≥ nS). Here, k2 is the rate constant of this polymerization, while the reverse process Pn + M ← Pn+1 does not play a significant role in this irreversible process. These two stages are described by the following kinetic equations with respect to the monomer concentration [M] and the concentration of ends of all stable polymers [P] (the concentration of ends is twice greater than the concentration of polymers): ⎧ d[P] = 2k+ns[M]n * ⎪ ⎪ dt ⎨ ⎪ d[M] = −k+ns[M]n * − k 2[P][M] ⎪ ⎩ dt
Figure 1. Visual representation of the lag-period, transition time, and relative size Lrel of the lag-period.
The Model of Linear Growth. The simplest model, which describes the formation of amyloid protofibrils, describes a straight, successive, and irreversible (after the nucleus formation) reaction, where the growth is initiated by the ends of polymers (Figure 2). Such a model has been already
(1)
In the latter equation of this system (first obtained by Oosawa et al.1), one can neglect the small term −k+nS[M]n* (i.e., the monomers spent to form a seed), which is permissible after the very beginning of the process (at the times t ≫ t0 = nS/(k2[MΣ])), when the fraction of monomers involved in the seeds is much smaller than the fraction of monomers involved in the long nascent polymers. System 1 can be strictly solved for the whole process of the reaction (see the Supporting Information, item 1): 41/ n * exp( −t /T ) [M] = [MΣ] [1 + exp( −n*t /T )]2/ n *
where T = (n*/(4k2k+nS[MΣ]n*))1/2 is a characteristic time of the descent of [M] at t → ∞, i.e., at the end of the polymerization reaction. In the case of linear growth, we found Lrel to be dependent only on the size n* of a nucleus and independent of the protein concentration, while durations of various stages of the linear growth are proportional to the initial monomer concentration in the power of −n*/2 (see the Supporting Information, eqs S.1.6−S.1.10, and Table 1). The Lrel(n*) function is presented numerically (Figure 3). As can be seen from Figure 3, Lrel is a monotonically decreasing function limited from the above, so that the experimental values of Lrel > 0.2 cannot describe the model of linear growth. However, the experimental data revealed Lrel to be more than 0.2 in many cases,4,22−25 which means that the linear growth regime model is not applicable in such cases and one needs to introduce scenarios where the number of protofibrils increases exponentially. First Model of Exponential Growth: the “Growth from the Surface” Scenario. The model describing the formation of amyloid protofibrils as an irreversible reaction, initiated by any monomer unit involved in the aggregate state, is shown in Figure 4. It should be noted that, in this scenario of amyloidogenesis, the fibrils are not favorable structures by definition; rather, a droplet-like structure will be formed. However, because certain experiments show some signs of this scenario (nonfibrillar β-structure rich aggregates),19 it is of potential interest to try to model it. A similar scheme of reaction was mentioned previously in a work of Morris et al.;12 however, no detailed analysis was done.
Figure 2. Free energy change during the initiation and linear growth of a protofibril. Squares: free monomer. Balls: monomer in an aggregated form. “Nucleus” is the most unstable aggregate at the reaction pathway, and “seeds” is the smallest stable aggregate.
introduced by Oosawa et al.,1 and an analytical solution for appropriate system of equations was provided. However, for our analysis, it is convenient to rewrite equations in terms of characteristic times which were introduced in the previous section; thus, the key values would be obtained more naturally. In the approximation given by the transition state theory,20,21 a quasi-stationary irreversible process of polymerization is described by two kinetic phases: (1) The irreversible initiation of polymerization, i.e., formation of the seeds ns M → PnS k+
(the reverse process PnS → nSM plays no significant role if the concentration of seeds, which are rapidly involved in further polymerization, is low). The seed consists of nS monomers M; nS is the smallest number of monomers in the stable polymer P. The process goes with an effective rate constant k+, which is defined by the height of the free-energy barrier, corresponding to the nucleus comprising n* monomers; n* < nS (see Figure 2); each polymer P has two ends capable of binding new monomers M. (2) The further irreversible polymerization of monomers M at the ends of already formed protofibrils P 1191
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Table 1. Theoretical Dependencies of Parameters of Experimental Kinetic Curves on the Total Concentration of Monomers and the Size of the Nuclei scenario of growth linear growth
growth from the surface
dependence of the characteristic duration of the aggregate growth (T2) and of the relative lag-period duration (Lrel) on the total concentration of monomers [MΣ]
eqs S.1.7−S.1.9 (SupportingInformation)
n* = − 2
Lrel = const ’ < 0.2
eq S.1.10 (SupportingInformation), Figure 3 eq S.2.8, eqs S.2.9 and S.2.7 (SupportingInformation)
n* = 2 − 4
ln T2 = const − ln[MΣ]
{
Lrel = const′ −
bifurcation
direct formula for calculation of n* and n2 through the slopes of the experimental lines
n* ln[MΣ] 2 n* ln Tlag = const − ln[MΣ] + ln{const′} 2 ln T2 = const −
ln Tlag = const − ln[MΣ] + ln const′ −
fragmentation
references on equations and figures
d(ln T2) d(ln[MΣ])
n* − 2 ln[MΣ] 4
}
d(Lrel) d(ln[MΣ])
eq 3, eq S.2.11 (SupportingInformation)
(n* − 2) ln[MΣ] 4
1 ln[MΣ] 2 1 ln Tlag = const − ln[MΣ] + ln{const′ − (n* − 1) ln[MΣ]} 2
eq S.3.3, eqs S.3.8 and S.3.9 (SupportingInformation)
n* = 1 −
Lrel = const′ − (n* − 1) ln[MΣ]
eq 5, eq S.3.10 (SupportingInformation) eq S.4.3, eqs S.4.4 and S.3.9 (SupportingInformation)
n2 = −1 − 2
ln T2 = const −
n2 + 1 ln[MΣ] 2 n +1 ln Tlag = const − 2 ln[MΣ] + ln{const′ − (n* − n2 − 1) ln[MΣ]} 2
ln T2 = const −
d(Lrel) d(ln[MΣ])
d(ln T2) d(ln[MΣ])
n* = 1 + n2 −
d(Lrel) d(ln[MΣ])
eq S.4.5 (SupportingInformation)
Lrel = const′ − (n* − n2 − 1) ln[MΣ]
(1) The initiation of polymerization, i.e., formation of the seed ns M → PnS k+
As above, PnS is a minimal stable unit (consisting of nS monomers M), which forms with the effective rate constant k+, determined by the height of the free-energy barrier (i.e., a “nucleus” of n* monomers; n* < nS). (2) The sum of all aggregations of monomers at polymers Pn + M ⎯→ ⎯ Pn + 1 nk 2
when n ≥ nS. It is the further irreversible aggregation at the already formed aggregates in which the reverse process Pn + M ← Pn+1 does not play a significant role. The speed of the process is proportional to the number [P]all = [MΣ] − [M] of monomers in all aggregates. As a result, we obtain
Figure 3. Numerical calculation of Lrel as a function of the nucleus size n* (see eq S.1.10, Supporting Information).
d[M] = −2k+ns[M]n * − k 2([MΣ] − [M]) ·[M] dt
(2)
Equation 2 can be solved for the early stages of the reaction, where [MΣ] − [M] ≪ [MΣ], and thus [M] ≈ [MΣ] (see the Supporting Information, item 2):
Figure 4. Dependence of the free energy on the reaction coordinates in the process of amyloidogenesis where the growth occurs at the surface of aggregates.
(1 + λ)e−xt [M] = [MΣ] 1 + λ ·e−xt
Here λ = k2/(2k+ns[MΣ]n*−2), x = k2[MΣ](1 + 1/λ), and x ≡ 1/ T2, where x and T2 are the rate and time of the exponential growth, correspondingly.
In the approximation given by the transition state theory,20,21 a quasi-stationary irreversible aggregation process is described by two kinetic phases: 1192
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Lrel can be easily found when λ ≫ 1, which means that the rate of initiation of aggregation (connected with k+) is much smaller than the rate of growth of the aggregates (connected with k2): Lrel ≈
⎛ ⎞ ⎫ k2 1⎜ ⎧ ⎬ − 2⎟⎟ ln⎨ ⎜ n *−2 4 ⎝ ⎩ 2k+ns[MΣ] ⎭ ⎠ ⎪
⎪
⎪
⎪
A protofibril divides into two stable parts; the chance of fragmentation for a single protofibril is roughly proportional to its length, lP, with the rate constant λ+. The possibility to merge two protofibrils into one (with the rate constant λ−) is not important at a low concentration of protofibrils at the beginning of the process. These stages are described by the following equations with respect to the concentration [M] of monomers M and the concentration [P] of ends of all the stable polymers P:
(3)
(see the Supporting Information, item 2). The important conclusion that follows is that there is no upper limit of Lrel (as it was in the case of the linear regime of growth, see Figure 3) when [MΣ] → 0 and n* > 2 or [MΣ] → ∞ and n* < 2, while at n* = 2 Lrel does not depend on [MΣ] (see Table 1). A similar conclusion (as it would be shown further) is applicable in the case of other forms of exponential regime of growth of amyloid protofibrils. Another Model of Exponential Growth: the “Fragmentation” Scenario. This model also describing the formation of amyloid as irreversible polymerization is shown in Figure 5. This scheme was first used for a description of the
⎧ d[P] = 2k+[M]n * + 2λ+([MΣ] − [M]) ⎪ d t ⎪ − λ−[P]2 ⎨ ⎪ d[M] ⎪ = −k+ns[M]n * − k 2[P][M] ⎪ ⎩ dt
(4) 5,27
In the latter equation of this system (cf. Ferrone et al. and Wegner26), one can neglect (as in system 1) the relatively small term −k+ns[M]n*; also, at the beginning of the process, [MΣ] − [M] ≪ [MΣ], and [P] is small; thus, λ−[P]2 can be neglected in the first equation, while [M] can be replaced by [MΣ] in k+[M]n* and in k2[P][M]. Then, ⎧ d[P] ≈ 2k+[MΣ]n * + 2λ+([MΣ] − [M]) ⎪ ⎪ dt ⎨ ⎪ d[M] ≈ −k 2[P][MΣ] ⎪ ⎩ dt
(4a)
for early stages of the reaction; solution of this system (see Supporting Information, item 3) gives
Figure 5. Dependence of the free energy of the reaction coordinates in the process of amyloidogenesis with fragmentation.
[M] = 1 − A(et / T2 + e−t / T2 − 2) [MΣ]
kinetics of actin filament formation;26 however, analytical solution was obtained only in a paper by Knowles et al. in 2009.10 Because of consideration of secondary nucleation and fragmentation cases together, the authors’ analysis lacked major relations, which could be used for determination of nucleus size, n*, so in this paper we will reconsider both cases separately. In the approximation given by the transition state theory,20,21 a quasi-stationary process of irreversible polymerization with fragmentation is described by three kinetic phases: (1) The initiation of polymerization (the same as in the reactions shown in Figures 3 and 4) ns M → PnS
where A = k+[MΣ]n*−1/2λ+ and T2 = 1/(4λ+k2[MΣ])1/2. We found L rel from this solution (see Supporting Information, item 3): ⎛ ⎞ 1 λ+ ⎟− Lrel = ln⎜⎜ n *−1 ⎟ 2 ⎝ k+[MΣ] ⎠
(5)
It is clearly seen that the dependence of Lrel on [MΣ] vanishes when n* = 1 (see Table 1). Moreover, in contrast to the model of the “growth from the surface scenario”, where Lrel decreases with an increase of [MΣ] when n* > 2, in the “fragmentation scenario”, Lrel decreases with an increase of [MΣ] when n* > 1, and a case where Lrel increases with an increase of [MΣ] is absent (because n* cannot be less than 1). One More Model of Exponential Growth: the “Bifurcation” Scenario. A model describing the formation of amyloid as a branching reaction, where the branching is initiated by the secondary nucleus attached to the inner parts of the polymer and the following growth occurs at the ends of the polymer and its branches, is shown in Figure 6. Initially, a similar model applied to biological matter was the description of hemoglobin aggregation during sickle cell anemia decease.4 However, an analytical solution was provided (as in a case of fragmentation scenario) in the paper by Knowles et al.;10 still, as it was mentioned above, the provided analysis lacked completeness, so in this paper we will reconsider the scheme. The process of irreversible polymerization with chain branching is described by three kinetic phases, only one of
k+
(2) The further irreversible (as in Figures 3 and 4) reaction of polymerization on the ends of the already formed protofibrils, where the reverse process Pn + M ← Pn+1 does not play a significant role P + M ⎯→ ⎯ P 2k 2
(or, more precisely, the sum of Pn + M ⎯→ ⎯ Pn + 1 2k 2
for all reactions with n ≥ nS). (3) A fragmentation of protofibrils λ−
P ←→ P + P λ+lP
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⎛ ⎞ 1 λ+′ ⎟− Lrel = ln⎜⎜ n *−n2 − 1 ⎟ 2 ⎝ 2k+[MΣ] ⎠
(8)
The dependence of Lrel on [MΣ] vanishes when n* − n2 = 1, and Lrel increases with an increase of [MΣ] when n* − n2 < 1 and decreases when n* − n2 > 1. It should be noted that the case of n2 = 0 is quite possible: this case means that the initiation energy is connected with deformation of the protofibril rather than with the deformation of the adhering monomer. This case will be experimentally investigated in our following work. Analysis of Concentration Dependencies. As it was mentioned above, the considered schemes of aggregation kinetics result in formulas by which one is able to calculate the size of the protofibril nucleus, n*, and (in the case of a bifurcation scenario) of the branched nucleus, n2. Such a calculation can be made through the analysis of the concentration dependences of two kinetic characteristics: the transition time (T2) and the ratio of duration of the lag-period to the transition time. These characteristics have a simple connection with the initial concentrations of monomers (see Table 1), and therefore, they are more appropriate for obtaining the sizes of primary and secondary nuclei than the often used (see, e.g., ref 10) duration of the lag-period itself. For these calculations, it is necessary to draw an array of points which reflects Lrel and ln(T2) for given molecules as a function of the initial monomer concentration [MΣ] (see Table 1 and the section “Transition Time and Size of the LagPeriod”). To calculate n* or n2, the obtained array should be approximated by a straight line using the method of leastsquares (see Figure 7). The error of the tangent coefficient is estimated as ε = tα0,(n−2)[(1/(n − 2))∑ni=1(yi − ŷi)2)/(∑ni=1(xi − x)̅ 2)]1/2,28 where n is the number of experimental [MΣ] values,
Figure 6. Dependence of the free energy on the reaction coordinates in the process of amyloidogenesis with bifurcations.
which is only slightly different from polymerization with fragmentation; compare Figures 5 and 6: (1) The initiation of polymerization (the same as in Figures 3−5) ns M → PnS k+
(2) The further irreversible (as in Figures 3−5) polymerization on the ends of already formed protofibrils and their branches Pn + M ⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 ends × k 2
(more precisely, the sum of all Pn + M ⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 ends × k 2
with n ≥ nS). (3) The emergence of a stable second protofibril on the first one (the speed of the emergence of such a protofibril is proportional to the length of the base protofibril, lP, and in total over all protofibrilsto the number of monomers in them)
P ⎯→ ⎯ P, P′ λ′+lP
Unlike the fragmentation process shown in Figure 5, which adds to the system two points of growth (two ends of the protofibril), the branching adds only one point of growth (end of the protofibril P′). The beginning of the branching process, when [MΣ] − [M] ≪ [MΣ] and [P], the concentration of ends of all the stable polymers, is also very small, is described by equations that are almost identical to those describing the beginning of the fragmentation (see eq 4a): ⎧ d[P] ≈ 2k+[MΣ]n * + λ+′ [MΣ]n2 ([MΣ] − [M]) ⎪ ⎪ dt ⎨ ⎪ d[M] ≈ −k 2[P][MΣ] ⎪ ⎩ dt
(6)
with the substitution 2λ+ ⇒ λ+′ [MΣ] , where λ+′ is no longer the rate constant of fragmentation but rather the bifurcation rate constant and n2 is the effective number of monomers in the nucleus of a branch. Therefore, the whole mathematical description of the beginning of the process of branching coincides with the description of the fragmentation process (see Supporting Information, item 3, and Table 1): n2
[M] = 1 − A(et / T2 + e−t / T2 − 2) [MΣ]
where A = k+[MΣ] while
n*−n2−1
/λ+′ and T2 = 1/(2λ+′ k2[MΣ]
Figure 7. ln T2 and Lrel versus ln[MΣ] for Yeast Prion Sup3523 with the approximation lines calculated by the least-squares method. The tangent coefficient kT2 = d(ln T2)/d(ln[MΣ]) of the approximation line is kT2 = −0.24 ± 0.45, and the tangent coefficient kLrel = d(Lrel)/ d(ln[MΣ]) of the approximation line is kLrel = −0.09 ± 0.20.
(7) n2+1
])1/2, 1194
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Table 2. The Range of Lrel and ln T2 Values and the Nuclei Sizes n*, n2 Calculated from the Experimental Data protein or peptide and a reference to the experimental study of the amyloid formation by this protein or peptide insulin22 17
Aβ42 Yeast Prion Sup3523 Yeast Prion Ure2p24 murine FBP28 WW domain25 β2-microglobulin29 27th immunoglobulin domain from human cardiac titin (TI I27)29 apolipoprotein C-II30
model that follows from the Lrel and T2 calculation
Lrel estimated at various ln[MΣ] (min − max)a
ln T2 estimated at various ln[MΣ] (min − max)
n* ± εn*, primary nucleusb
n2 ± εn2, secondary nucleusb
exponential (fragmentation/ bifurcation) bifurcation exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) exponential (fragmentation/ bifurcation) bifurcation
5.17−5.56
−0.49−0.08
1.13 ± 0.19
−0.48 ± 0.16
0.53−0.67 0.29−0.70
−0.51−1.44 4.62−5.52
2.64 ± 0.11 1.09 ± 0.20
1.72 ± 0.05 −0.51 ± 0.45
0.76−1.02
1.79−2.39
0.96 ± 1.52
−0.23 ± 1.40
1.56−2.10
4.76−6.02
1.21 ± 1.27
0.05 ± 1.02
1.48−3.86
0.79−2.28
1.58 ± 0.58
−0.06 ± 0.18
0.14−0.34
5.93−7.85
2.86 ± 0.30
2.04 ± 0.29
0.06−0.10
2.77−4.82
linear
a
The errors are computed using the Student’s coefficient corresponding to 0.95 confidence level. amyloidogenesis kinetics were kindly provided by S. E. Radford.
yi is the value of Lrel (or ln T2) corresponding to the ith value of [MΣ], ŷi is the approximated value of yi, xi̅ ≡ [MΣ]i, x̅ is the averaged [MΣ]i value, and tα0,(n−2) is the Student’s coefficient with confidence α0 = 0.95 and n − 2 degrees of freedom.
b
4.44 ± 0.38
Numerical data for β2-microglobulin
In ref 10, it was noted that most of the experimental data in the appropriate coordinates (with our definitions - (ln Tlag, ln[MΣ])) lies close to the line ln Tlag ∼ −(1/2) ln[MΣ] + const. Moreover, almost all studied proteins had Lrel ≫ 0.2, which means that their aggregates had an exponential mechanism of growth. This intriguing feature compelled us to take the existing experimental data on amyloid formation kinetics (see Table 1) and analyze them with our model. Table 1 suggested that a linear dependence ln Tlag ∼ −(1/2) ln[MΣ] + const can correspond only to the cases of the fragmentation or bifurcation scenarios with n* = 1 and n2 = 0. The same conclusion is obtained from the analysis of Lrel values as a function of MΣ. For the proteins insulin,22 Sup35,23 Ure2p,24 WW,25 Aβpeptide,17 and β2-microglobulin,29 it is definitely an exponential scenario of protofibril growth (Lrel ≫ 0.2). Moreover, the determination of n* directly from the experimental data in proposed coordinates (Lrel, ln[MΣ]) gives n* ≈ 1 (within the error) for all of these proteins (except for Aβ-peptide) with an exponential mechanism of growth (see Table 2). Aβ-peptide was the only object among those mentioned above which directly points toward the bifurcation scenario of growth having the primary nucleus size equal to n* ≈ 3 and the secondary nucleus size equal to n2 ≈ 2. In all the other cases, the secondary nucleus size is close to 0 (within errors), with a possible exception of insulin, where the small negative value of n2 may mean that some protofibrils lose monomers in the process of fragmentation or bifurcation. Thus, our results are in reasonable accordance with those of the authors whose data we used.17 Among all proteins used in ref 10 (and listed in Table 2), TI I2729 protein and apolipoprotein C-II30 got very low Lrel (0.25 ± 0.12 and 0.07 ± 0.03), which is typical of the linear growth scenario. Calculation of the nucleus size for apolipoprotein CII30 gives n* ≈ 4. However, the concentration dependence of Lrel on [M] for TI cannot be explained in terms of the linear growth scenario, so we applied an exponential mechanism analysis for this protein. Calculation showed the bifurcation scenario to be an adequate candidate for approximation of the experimental data. The corresponding dependences (ln T2, ln[MΣ]) and (Lrel, ln[MΣ]) give the primary nucleus size n* ≈ 2 and secondary nucleus size n2 ≈ 3, respectively.
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RESULTS AND DISCUSSION All the obtained dependencies are summarized in Table 1, and the procedure of determining what mechanism of amyloid formation is present in a given experiment should be done in the following way. If Lrel > 0.2 independently on [MΣ] the model of amyloid protofibril formation through the linear regime of growth is not applicable, and it is a clear case of the exponential mechanism of growth. Further determination of what exact scenario of exponential mechanism of growth is present in a given experiment can be done on the basis of the dependence of ln T2 from ln[MΣ] (see Table 1). The dependence of ln T2 on ln[MΣ] is different for growth from the surface scenario, for fragmentation, and for bifurcation scenarios. However, it is impossible to distinguish the bifurcation from fragmentation scenarios from kinetics, because all the kinetic dependencies for them have identical forms, if the size of the secondary nucleus is equal to zero. This statement implies the need for direct experimental data to determine if it is a fragmentation or bifurcation scenario. As it can be seen from Table 1, all the predicted dependencies of Lrel and ln T2 on ln[MΣ] are linear (unlike the ln Tlag on ln[MΣ] dependence), which provides a convenient possibility for calculation of the sizes n* and n2 of primary and secondary nuclei from the kinetic experimental data presented in coordinates (ln T2, ln[MΣ]) and (Lrel, ln[MΣ]) using the least-squares method. It should be reiterated that, for simplicity, we assumed that a “monomer” floating in solution includes only one protein molecule; if the floating “monomers” consist of nmono protein molecules (which can be established only experimentally), the values n* and n2 (which give the size of the nuclei in the number of “monomers”) must be multiplied by nmono to obtain the size of the nuclei in the numbers of protein molecules. In the case of exponential growth, it is better to use the Lrel dependence on ln[MΣ] and the ln T2 dependence on ln[MΣ] instead of ln Tlag on ln[MΣ] (used in ref 10), because in the first case the dependence has a linear form for every scenario of exponential growth. 1195
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(7) Cali, I.; Castellani, R.; Yuan, J.; Al-Shekhlee, A.; Cohen, M. L.; Xiao, X.; Moleres, F. J.; Parchi, P.; Zou, W. Q.; Gambetti, P. Classification of Sporadic Creutzfeldt-Jakob Disease Revisited. Brain 2006, 129, 2266−2277. (8) Serio, T. R.; Cashikar, A. G.; Kowal, A. S.; Sawicki, G. J.; Moslehi, J. J.; Serpell, L.; Arnsdorf, M. F.; Lindquist, S. L. Nucleated Conformational Conversion and the Replication of Conformational Information by a Prion Determinant. Science 2000, 289, 1317−1321. (9) Xue, W. F.; Homans, S. W.; Radford, S. E. Systematic Analysis of Nucleation-Dependent Polymerization Reveals New Insights into the Mechanism of Amyloid Self-Assembly. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 8926−8931. (10) Knowles, T. P.; Waudby, C. A.; Devlin, G. L.; Cohen, S. I.; Aguzzi, A.; Vendruscolo, M.; Terentjev, E. M.; Welland, M. E.; Dobson, C. M. An Analytical Solution to the Kinetics of Breakable Filament Assembly. Science 2009, 326, 1533−1537. (11) Cohen, S. I.; Vendruscolo, M.; Welland, M. E.; Dobson, C. M.; Terentjev, E. M.; Knowles, T. P. Nucleated Polymerization with Secondary Pathways. I. Time Evolution of the Principal Moments. J. Chem. Phys. 2011, 135, 065105. (12) Morris, A. M.; Watzky, M. A.; Finke, R. G. Protein Aggregation Kinetics, Mechanism, and Curve-Fitting: A Review of the Literature. Biochim. Biophys. Acta 2009, 1794, 375−397. (13) Giehm, L.; Otzen, D. E. Strategies to Increase the Reproducibility of Protein Fibrillization in Plate Reader Assays. Anal. Biochem. 2010, 400, 270−281. (14) Masel, J.; Jansen, V. A.; Nowak, M. A. Quantifying the Kinetic Parameters of Prion Replication. Biophys. Chem. 1999, 77, 139−152. (15) Loksztejn, A.; Dzwolak, W. Vortex-Induced Formation of Insulin Amyloid Superstructures Probed by Time-Lapse Atomic Force Microscopy and Circular Dichroism Spectroscopy. J. Mol. Biol. 2010, 395, 643−655. (16) Andersen, C. B.; Yagi, H.; Manno, M.; Martorana, V.; Ban, T.; Christiansen, G.; Otzen, D. E.; Goto, Y.; Rischel, C. Branching in Amyloid Fibril Growth. Biophys. J. 2009, 96, 1529−1536. (17) Cohen, S. I.; Linse, S.; Luheshi, L. M.; Hellstrand, E.; White, D. A.; Rajah, L.; Otzen, D. E.; Vendruscolo, M.; Dobson, C. M.; Knowles, T. P. Proliferation of Amyloid-β42 Aggregates Occurs through a Secondary Nucleation Mechanism. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 9758−9763. (18) Jeong, J. S.; Ansaloni, A.; Mezzenga, R.; Lashuel, H. A.; Dietler, G. Novel Mechanistic Insight into the Molecular Basis of Amyloid Polymorphism and Secondary Nucleation During Amyloid Formation. J. Mol. Biol. 2013, 425, 1765−1781. (19) Cho, K. R.; Huang, Y.; Yu, S.; Yin, S.; Plomp, M.; Qiu, S. R.; Lakshminarayanan, R.; Moradian-Oldak, J.; Sy, M.-S.; De Yoreo, J. J. A Multistage Pathway for Human Prion Protein Aggregation in Vitro: From Multimeric Seeds to β Oligomers and Nonfibrillar Structures. J. Am. Chem. Soc. 2011, 133, 8586−8593. (20) Eyring, H. The Activated Complex in Chemical Reactions. J. Chem. Phys. 1935, 3, 107−115. (21) Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; Wiley: Chichester, England 1981. (22) Fodera, V.; Librizzi, F.; Groenning, M.; van de Weert, M.; Leone, M. Secondary Nucleation and Accessible Surface in Insulin Amyloid Fibril Formation. J. Phys. Chem. B 2008, 112, 3853−3858. (23) Collins, S. R.; Douglass, A.; Vale, R. D.; Weissman, J. S. Mechanism of Prion Propagation: Amyloid Growth Occurs by Monomer Addition. PLoS Biol. 2004, 2, 1582−1590. (24) Zhu, L.; Zhang, X. J.; Wang, L. Y.; Zhou, J. M.; Perrett, S. Relationship between Stability of Folding Intermediates and Amyloid Formation for the Yeast Prion Ure2p: A Quantitative Analysis of the Effects of pH and Buffer System. J. Mol. Biol. 2003, 328, 235−254. (25) Ferguson, N.; Berriman, J.; Petrovich, M.; Sharpe, T. D.; Finch, J. T.; Fersht, A. R. Rapid Amyloid Fiber Formation from the FastFolding WW Domain Fbp28. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 9814−9819. (26) Wegner, A. Spontaneous Fragmentation of Actin Filaments in Physiological Conditions. Nature 1982, 296, 266−267.
CONCLUSIONS We have shown that kinetic curves can give insight into various mechanisms of amyloid aggregate formation. The first indicator of a possible regime of growth can be derived from Lrel, where the concentration-independent Lrel ≈ 0.2 is the border between linear and exponential scenarios of protofibril growth. Some conclusion can also be made about a kind of the exponential growth scenario which presents the given experiment (e.g., growth from the surface or fragmentation/bifurcation scenario; the distinction of the two latter scenarios requires direct approaches (for example, real-time observations of growing protofibrils at a single-protofibril level, as in ref 16)). The suggested theory of concentration-dependent kinetics can also reveal such important parameters of the reaction as the sizes of the primary and secondary folding nuclei of protofibrils. In the very end, we should add that the above given models correspond to “pure” cases of the amyloid formation via different mechanisms, while a mixture of several mechanisms can be observed sometimes in experiments.18
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ASSOCIATED CONTENT
S Supporting Information *
Solution of systems of kinetic equations for all discussed aggregation scenarios. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Phone/fax: +74967-318275/ +74967-318435. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are grateful to Prof. S. E. Radford who kindly provided us with numerical data for β2-microglobulin amyloidogenesis kinetics. We thank T. B. Kuvshinkina for assistance in the manuscript preparation. This study was supported in part by the Russian Foundation for Basic Research (Grant Nos. 11-0400763a and 13-04-00253a), Russian Academy of Sciences (programs “Molecular and Cell Biology” (Grant Nos. 01201353567 and 01201358029) and “Fundamental Sciences to Medicine”).
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