Modeling the Effect of Monomer Conformational Change on the Early

Article pubs.acs.org/JPCB

Modeling the Effect of Monomer Conformational Change on the Early Stage of Protein Self-Assembly into Fibrils Dimo Kashchiev* Institute of Physical Chemistry, Bulgarian Academy of Sciences, ul. Acad. G. Bonchev 11, Sofia 1113, Bulgaria S Supporting Information *

ABSTRACT: Filamentous self-assembly of proteins is an important process implicated in a plethora of human diseases and of interest for nanotechnology. Using rate equations, we analyze the early stage of the process in solutions that initially contain fibrillation-passive protein monomers and in which the nascent fibrils are practically insoluble. The analysis is based on a model accounting for the conformational and/or other changes the passive monomers experience to transform themselves into fibrillation-active monomers and thus become fibril nuclei. The model allows exact, comprehensive, and simple mathematical description of the early stage of fibrillation, which reveals the usually neglected role of the nucleation nonstationarity in this stage of fibrillation. We obtain exact and user-friendly expressions for experimentally accessible quantities such as the size distribution of fibrils, their number and mass concentrations, the rate and nonstationary period of fibril nucleation, and the delay time of fibril formation. Analyzing available experimental data, we find that the theory successfully describes the fibrillation time course of pathological and nonpathological ataxin-3, a protein involved in the neurodegenerative disorder spinocerebellar ataxia type-3. The analysis provides mechanistic insight into the reason for the higher fibril nucleation and elongation rates of the pathological ataxin-3.

■

INTRODUCTION Filamentous self-assembly of proteins in solutions is an important process involved in scores of human diseases and of interest for nanotechnology. This process is a particular case of formation of new condensed phases and thus has much in common with the formation of droplets or crystallites in solutions. For example, initially, all of the fibrils, droplets, and crystallites usually grow by random attachment and detachment of monomeric solute to and from them. This similarity in their kinetic behavior has made it possible for Oosawa and Kasai1 to pioneer the rate equations approach of nucleation theory2,3 for describing the kinetics of protein fibrillation. Heretofore, rate equations have been widely used to study not only the early but also the advanced stage of fibrillation, which may be kinetically quite complicated because of secondary processes such as fibril fragmentation, branching, and lateral association (see, e.g., refs 4−28 and recent reviews29,30). In the rate equations approach to fibrillation, it is usually presumed that already at the onset of the process the protein monomers in the solution are fibrillation-active because of being in conformational state (e.g., misfolded), allowing them to polymerize into fibrils. Much fewer are the studies that use rate equations accounting for the possibility for the monomers to be in conformational state (e.g., natively folded) that prevents them from assembling into fibrils and thus renders them fibrillation-passive. Transformation of the protein monomers from fibrillation-passive into fibrillation-active ones (called hereafter P monomers and A monomers, respectively) © XXXX American Chemical Society

occurs for various reasons (see, e.g., ref 31). This transformation may be an important prefibrillation step for solutions populated by P-monomers19,22,28,32−45 and an essential determinant in the fibril elongation rate.46 It has also been shown47−52 that this transformation can crucially affect the pathways of fibril formation and thereby the entire protein fibrillation kinetics. Factors for the monomers to be fibrillationpassive are, for instance, low enough temperature35,53,54 or pressure35 and absence or presence of various ions,5,55,56 substances,37,38 or nanoparticles57,58 in the protein solution. Significantly, being fibrillation-passive, the protein monomers may still aggregate before their activation but into disordered, possibly disease-related oligomers, which can be either on or off the pathway of fibrillation. This protein aggregation scenario resembles that in ref 59 and is beyond the scope of the present study. To account for the effect of monomer conformational change, the Oosawa−Kasai (OK) model1 and models more or less related to it were expanded by inclusion of a monomer activation step prior to polymerization, and the rate equations describing the models were modified accordingly (see, e.g., refs 5, 9, 10, 15, 17, 19, 22, 27, and 28). It was found inter alia that slow enough monomer activation can considerably change the lag time and the maximum rate of fibrillation.5 Importantly, Received: September 14, 2016 Revised: November 15, 2016

A

DOI: 10.1021/acs.jpcb.6b09302 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

can attach P monomers to their two ends and in the process transform them into A monomers. The attachment occurs with frequency kZ, where k (M−1 s−1) is the n-independent Pmonomer attachment frequency per unit P-monomer concentration. Detachment of A monomers from the fibril ends is impossible because the fibrils are insoluble, and the fibrils are not subject to fragmentation, association, or other processes that may occur after the early stage of fibrillation. The fibril elongation rate Re ≡ dn/dt is thus simply given by Re = kZ. Also, because the fibrils are linear chains of A monomers differing in length only, there is no need for specific terms for the short and the long protein chains, such as the often used protofibril and mature fibril, respectively. The model schematized in Figure 1a and analyzed below is the Chen−Ferrone−Wetzel (CFW) model34,39 for insoluble linear fibrils. It is both a modification and an extension of the OK model1 of linear polymerization of protein into insoluble fibrils (Figure 1b). The modification is that in the CFW model the fibril elongation is due to attachment of P monomers rather than of A monomers, as it is in the OK model. The extension is in the inclusion of the monomer activation step, which is an indispensable prefibrillation step when fibrils are to come into being in solutions initially populated by P monomers only. The CFW model (Figure 1a) parallels the Frieden−Goddette (FG) model5 for insoluble linear fibrils (Figure 1c), which also allows for monomer activation, but according to which, like in the OK model, A monomers and fibrils can attach solely A monomers to themselves. In fact, the CFW, OK, and FG models in Figure 1a−c are three particular cases of the unified model schematized in Figure 1d. This model requires a separate study because in it the fibrils attach both P and A monomers, the latter process occurring with frequency kAC1, where kA is the n-independent A-monomer attachment frequency per unit A-monomer concentration. The rate equations of the unified model for insoluble linear fibrils are presented by eqs S51 and S52 in the Supporting Information (SI). While in the case of kA = 0 these equations turn into eqs 1 and 2 (given below) of the CFW model in Figure 1a, in the a = b = k = 0 case they pass into the equations of the OK model1 (Figure 1b), and in the case of k = 0 these equations become the equations of the FG model5 (Figure 1c). Also, when monomer detachment from the fibrils is taken into account in the rate equations (see SI) the unified model becomes applicable to soluble linear fibrils. Whereas in the models of protein polymerization commencing with monomer-activation step the fibrils are usually considered as elongating by attaching A monomers, in the CFW model the fibril elongation is due to P-monomer attachment. Clearly, whether P or A monomers prevail in contributing to the fibril evolution depends on their concentrations in the solution, because while the former attach themselves to the fibrils with frequency kZ, the latter do so with frequency kAC1. Therefore, the CFW model is applicable when kAC1 ≪ kZ or, more simply, when C1 ≪ Z if kA is equal to or not too much different from k. In the OK model of linear fibrils,1 the growth of the A monomers into dimers, trimers, and so on is an energetically downhill process.60−62 These monomers can thus be regarded as the fibril nuclei, but only formally, because they are in the solution right at the onset of fibrillation. The A monomers in the CFW model34,39 also grow downhill but in contrast have to be born out of the P monomers if fibrillation in an A-monomerfree solution is ever to begin. In this sense, they are bona fide fibril nuclei.45 Hence, as in the classical nucleation theory,3 we

accounting for the monomer activation provides a rationale for conceiving the A-monomer as fibril nucleus when this monomer is the highest-energy species along the P monomer → A monomer → dimer → trimer → etc. pathway of fibril formation.19,22,28,33,34,36,38,39,41,43−45 The objective of this study is to describe the early stage of protein fibrillation in solutions that initially contain Pmonomers only and in which the nascent fibrils are practically insoluble. The description is based on a model that takes into account the monomer activation step during which the Pmonomers convert into A-monomers and thus become fibril nuclei. The goal is to obtain exact expressions for experimentally accessible quantities such as the size distribution of fibrils, their number and mass concentrations, and the rate and nonstationary period of fibril nucleation as well as to employ some of these expressions for analysis of existing experimental data for the time course of protein fibrillation.

■

THEORETICAL METHODS Model. The considered model of linear fibril formation is schematized in Figure 1a. We have a solution of fibrillogenic

Figure 1. Schematic of the Chen−Ferrone−Wetzel34,39 (a), Oosawa− Kasai1 (b), Frieden−Goddette5 (c), and unified (d) models of protein self-assembly into insoluble linear fibrils. (The circles and the rectangles represent P monomers and A monomers, respectively.)

protein monomers and insoluble linear fibrils. The monomers are of two kinds. The much more numerous of them, because of being in their native conformation or for some other reason, are passive with respect to self-assembling into either fibrils or nonfibrillar aggregates. These are the P-monomers (the circles in Figure 1), and their concentration is denoted by Z. The remaining monomers (the rectangles in Figure 1), with concentration C1 ≪ Z, are the A monomers. They are fibrillation-active because they are misfolded or are in another appropriate state allowing them to attach P monomers and become fibril dimers. The P and A monomers in the solution can convert into each other: Conversion of a P monomer into an A monomer occurs with frequency a (s−1), and an A monomer can convert back into a P monomer with frequency b (s−1). The fibrils of size n (see Figure 1) and concentration Cn at time t are constituted of n linearly arranged A monomers (n = 2, 3, 4, ..., ad infinitum) because like the A monomers they B


Article

The Journal of Physical Chemistry B can speak of fibril nucleation rate, meaning the frequency with which the fibril nuclei (the A-monomers) become the fibrils of one more monomer (the dimers). However, hereafter using this and other terms from nucleation theory, we shall always bear in mind that the birth of the fibril nucleus, the process of P monomer converting into A monomer, is quite different from the classical nucleation process. Also, having the monomer as the fibril nucleus, the CFW model in Figure 1a cannot describe fibrillation characterized by nucleus of two, three, or more monomers. Rate Equations. The rate equations describing protein fibrillation within the three-parameter CFW model in Figure 1a read (n = 2, 3, 4, ...) dC1 = aZ − (b + kZ)C1 dt

dCn = kZCn − 1 − kZCn dt

Cst =

(4)

is the stationary A-monomer concentration established in the t ≫ 1/(b + kZ) limit. Then, using eq 2 at successive values of n from n = 2 on, we find C2 with the aid of C1 from eq 3, C3 with the aid of the so-found C2, and so on (see SI). With this modus operandi, by mathematical induction, we finally arrive at the following exact expression for the sought fibril size distribution (n = 2, 3, 4, ...) ⎛ kZ ⎞n − 1 Cn(t ) = Cst − Cst⎜ − ⎟ e−(b + kZ)t ⎝ b ⎠ n−2 ⎡ j ⎛ ⎞n − j − 1⎤ −kZt − Cst e ∑ ⎢1 − ⎜⎝− kZ ⎟⎠ ⎥ (kZt ) b ⎦ j! j=0 ⎣

(1)

(5)

Substitution in eq 2 proves that this expression is indeed the exact solution of eq 2. As seen from eq 5, Cn(t) = Cst in the limit of t → ∞, so that Cst is also the stationary concentration of the fibrils of any size n. This concentration is thus of great physical significance because, given the Z value, via the three parameters a, b, and k of the CFW model, it sets up the upper limit of the concentrations of both the A monomers and the fibrils in the solution. For instance, when P-monomer activation is arrested (then a = 0) or A-monomer passivation is infinitely fast (then b = ∞), Cst from eq 4 vanishes and fibrillation is impossible. The solid lines in Figure 2a display the Cst(Z) dependence from eq

(2)

In these equations, the aZ and −bC1 terms are the rates of Amonomer gain and loss by conversion from passive into active and from active into passive states, respectively, the −kZC1 term is the rate at which the A monomers disappear when becoming dimers, and the kZCn−1 and −kZCn terms represent the rates of n-sized fibril appearance and disappearance by Pmonomer attachment to the (n − 1)-sized and the n-sized fibrils, respectively. (All of these rates are visualized by the arrows in Figure 1a.) Equations 1 and 2 are an infinite set of ordinary differential equations for the A-monomer concentration C1(t) and the fibril size distribution Cn(t). The problem of finding C1 and Cn is mathematically well-posed when this set is supplemented with an equation for the P-monomer concentration Z(t) and when the initial concentrations Z(0) and Cn(0) for n = 1, 2, 3, ... are specified. In fibrillation at a given time-independent value of Z, an equation for this quantity is not needed, and eqs 1 and 2 are linear and have the exact and rather simple solution given below. This solution, which is one of the main results of the present study, is applicable to protein fibrillation when the initial P-monomer concentration is kept fixed by some regulatory process. To a certain approximation, however, this solution also applies to the earliest stage of fibrillation at fixed total concentration Ctot of protein because during this stage, despite the loss of the P monomers by their attaching to fibrils, Z decreases less than several percent from its initial value. In this case of fibrillation, the necessary equation for Z is the equation Z(t) + ∑∞ n=1nCn(t) = Ctot for conservation of the total protein mass, which is why Z is timedependent and eqs 1 and 2 are nonlinear and difficult to treat analytically.

Figure 2. Dependence of (a) the stationary A-monomer concentration and (b) the stationary fibril nucleation rate on the P-monomer concentration at monomer activation frequency a = 0.001, 0.005, or 0.01 s−1 (as indicated): solid lines in panels a and b, eqs 4 and 9, respectively; dashed lines in panels a and b, linear and quadratic smallZ approximations, respectively.

■

4 at a = 0.001, 0.005, or 0.01 s−1 (as indicated), b = 1 s−1 and k = 104 M−1 s−1. We observe that when Z ≪ b/k, Cst increases linearly with Z and is practically equal to the equilibrium Amonomer concentration Ceq = aZ/b, whose Z dependence is visualized by the dashed lines in Figure 2a. This concentration would exist in the protein solution in the absence of fibrillation (then k = 0, and Ceq is the t-independent solution of eq 1). With increasing Z, however, Cst gradually falls below Ceq until assuming the plateau value of a/k when Z ≫ b/k. Exhibited in Figure 3 by solid lines is the time dependence of the concentration Cn of A monomers (n = 1) and fibrils of size n = 2, 3, 4, 6, 8, 10, and 12 (as indicated). The lines are drawn according to eqs 3 and 5 with a = 0.01 s−1, b = 1 s−1, k = 104

RESULTS AND DISCUSSION Fibril Size Distribution. Our task now is to obtain the solution Cn(t) of eqs 1 and 2 under initial condition Cn(0) = 0 for n = 1, 2, 3, ... in the case of t-independent a, b, k, and Z, the latter being the experimentally controllable initial value of the P-monomer concentration. First, from eq 1, we find that the exact time dependence of the A-monomer concentration is of the form C1(t ) = Cst[1 − e−(b + kZ)t ]

aZ b + kZ

(3)

where Cst, given by C


Article


1a these aggregates are the fibril dimers, J2(t) is the nonstationary nucleation rate characterizing the fibrillation process described by the model. Combining eq 7 at n = 2 with eqs 3 and 4, we readily find that J2 is of the form J2 (t ) = Jst (1 − e−t / τ )

(8)

where Jst and τ, given by Jst =

τ= Figure 3. Time dependence of the concentration of A-monomers (line 1) and of fibrils of size n = 2, 3, 4, 6, 8, 10, and 12 (as indicated): solid lines, eqs 3 and 5; dashed lines, eq 6.

akn − 1Z n n t n!

(6)

With the above parameter values, Cn(t) from this equation is illustrated by the dashed lines in Figure 3. Again with the above parameter values, the fibril size distribution from eqs 3 and 5 is plotted in Figure 4 but at

1 b + kZ

Jn (t ) =

(10)

J R en − 2 n − 1 akn − 1Z n n − 1 = st t t (n − 1)! (n − 1) ! τ

(11)

We note also that when kZ/b ≪ 1, Jn is given approximately by eq S19 in SI. Looking back at the stationary nucleation rate Jst from eq 9, we see that this physically important quantity depends on all three frequencies a, b, and k and that it increases quadratically with the P-monomer concentration Z when this is low (Jst = (ak/b)Z2 for Z ≪ b/k). When Z is high, however, Jst is linear with Z (Jst = aZ for Z ≫ b/k) and, notably, independent of the passivation and attachment frequencies b and k. In this regime, therefore, the stationary fibril nucleation is controlled solely by the activation frequency a so that an experimentally obtained linear Jst(Z) dependence allows determination of this frequency from the slope of the straight line. The solid lines in Figure 2b display Jst from eq 9 as a function of Z at a = 0.001, 0.005, or 0.01 s−1 (as indicated), b = 1 s−1, and k = 104 M−1 s−1. The dashed lines visualize the corresponding quadratic Jst(Z) dependence Jst = (ak/b)Z2, which is valid for Z ≪ b/k. The nonstationary period τ of nucleation is a quantity of particular interest because it may have a sizable contribution to the fibrillation lag time tlag often reported in experiments performed at fixed total protein concentration. In theoretical studies on fibrillation kinetics, it is usually assumed that the fibril nuclei are in pre-equilibrium with the subnucleus species in the solution. For the CFW model, this assumption is equivalent to the assumption that the fibril nucleation is stationary, that is, that τ is vanishingly small. When the passivation and attachment frequencies b and k are rather low, however, and when the P-monomer concentration Z is also low, τ may be minutes, hours, or even much longer and thus comparable to experimentally obtained fibrillation lag times tlag. This means that then the pre-equilibrium assumption is not realistic. For instance, with b = 10−4 s−1 and k = 102 M−1 s−1, eq

Figure 4. Fibril size distribution at times t = 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.5, and 2 min (as indicated): solid lines, eqs 3 and 5; dashed line, the stationary concentration Cst of A-monomers and fibrils.

successive times t = 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.5, and 2 min (as indicated). We see that after an initial transient period the size distribution develops a sigmoidal front that moves unrestrictedly toward larger fibril sizes. We note also that when kZ/ b ≪ 1, the kZ/b terms in eq 5 are negligible and Cn is given approximately by the simpler eq S11 in SI. Fibril Formation Rate. By definition, the fibril formation rate Jn is the frequency with which fibrils of a given size n appear in unit solution volume. In the three-parameter CFW model, Jn is thus merely the flux kZCn−1 from point n − 1 to point n on the fibril size axis, that is (n = 2, 3, 4, ...) Jn (t ) = kZCn − 1(t )

(9)

are, respectively, the stationary nucleation rate (established when t ≫ τ) and the nonstationary period of nucleation. Comparing eqs 3 and 8, we see that the latter follows from the former upon replacement of C1 by J2 and of Cst by Jst. Therefore, with Cn and Cst replaced by Jn+1 and Jst, respectively, eq 5 is the exact expression for the fibril formation rate Jn+1 at n = 2, 3, 4, ... (see also eq S14 in SI). Like the nucleation rate J2, all fibril formation rates J3, J4, and so on are nonstationary and have the same steady-state value Jst at long enough time. Because of eq 7, the time course of Jn is identical to that of Cn−1 (cf. Figure 3 and Figure S2 in SI). From eqs 6 and 7, it follows that Jn is initially a power function of time, independent of the monomer passivation frequency b provided this is finite (n = 2, 3, 4, ...)

M−1 s−1, and Z = 10 μM. As seen, with n-dependent delay, the concentration of the A monomers and all fibrils ultimately plateaus at Cst, whose value in the exemplified case is 0.09 μM. The Cn(t) function is sigmoidal for all n ≥ 2. As shown in SI, initially Cn rises as tn and is independent of the passivation frequency b provided this frequency is finite (n = 1, 2, 3, ...) Cn(t ) =

akZ2 b + kZ

(7)

In nucleation theory, the nucleation rate is defined as the formation rate of the aggregates containing one more monomer than the nucleus.3 Hence, because in the CFW model in Figure D


Article

The Journal of Physical Chemistry B 10 yields τ = 1.4 h and 1.7 min at Z = 1 and 100 μM, respectively. The τ(Z) dependence from eq 10 is displayed in Figure 5a, with the lines being drawn with b = 10−4, 0.1, 1, or 5

N2(t ) = Jst [t − τ(1 − e−t / τ )]

(13)

⎡ ⎤ ⎛ kZ ⎞n − 2 Nn(t ) = Jst ⎢t − θn + τ ⎜ − ⎟ e−t / τ ⎥ ⎝ b ⎠ ⎣ ⎦ + Jst

e−kZt kZ

⎧⎡

j=0

⎪

⎛

⎞n − j − 2 ⎤ ⎥ b ⎦

⎪ ⎩⎣

i⎪ ⎫

j

×

n−3

∑ ⎨⎢1 − ⎝⎜− kZ ⎠⎟

∑ (kZt ) ⎬ i=0

i! ⎪ ⎭

(14)

which are eqs S23 and S25 in SI and in which Jst, τ, and θn are specified by eq 9, eq 10, and the equation (n = 2, 3, 4, ...) θn =

1 n−2 n−2 + =τ+ b + kZ kZ Re

(15)

For example, at n = 4, eq 14 leads to eq S28 in SI for the number concentration N4 of tetramers and longer fibrils. It follows from eqs 13 and 14 that when fibrillation is so advanced that t ≫ θn, like in nucleation theory,3 Nn is a linear function of time given by the equation (n = 2, 3, 4, ...)

Figure 5. Dependence of (a) the nonstationary period of nucleation and (b) the delay time of tetramer formation on the P-monomer concentration at monomer passivation frequency b = 10−4, 0.1, 1, or 5 s−1 (as indicated): lines in panels a and b, eqs 10 and 15, respectively.

s−1 (as indicated) and with k = 104 M−1 s−1. We observe a monotonic decrease of τ with increasing Z, which is similar to that of experimentally reported fibrillation lag times tlag. It is important to note as well that according to eq 10 τ is independent of the monomer activation frequency a. Also, as Re = kZ, it follows from eq 10 that τ < 1/Re. This useful relation sets up an upper limit of the period τ of nucleation nonstationarity: For example, if the fibrils elongate at rate Re = 1 min−1, then this period cannot be longer than 1 min. Another useful relation, Cst = Jst/Re, which follows from eqs 4 and 9 or from eq 7, allows calculation of the stationary Amonomer concentration Cst with the aid of experimentally available rates Re and Jst of fibril elongation and stationary nucleation. Fibril Number Concentration. Knowing the fibril size distribution Cn, we can now determine the experimentally accessible number concentration Nn of fibrils of size n, n + 1, n + 2, and so on. As Nn is defined by Nn(t) = ∑∞ j=nCj(t), N2 represents the number concentration of all fibrils in the solution. However, obtaining N2 is not always possible experimentally because it requires the use of a technique allowing detection even of the fibril dimers. In experiments, often the smallest detectable fibril size may not be 2 but a considerably greater integer. For that reason, it is important to know Nn for any n ≥ 2. To find the Nn(t) dependence corresponding to the threeparameter CFW model, rather than summing Cn from eq 5 in accordance with the above definition of Nn, we shall use the equivalent expression (n = 2, 3, 4, ...)

Nn(t ) = Jst (t − θn)

(16)

which shows that, physically, θn is the delay time of formation of n-sized fibrils. In the opposite case of t ≪ θn, that is, early at the onset of fibrillation, provided the monomer passivation frequency b is finite, Nn is independent of this frequency and increases parabolically with time (n = 2, 3, 4, ...) Nn(t ) =

J R en − 2 n akn − 1Z n n t = st t n! n!τ

(17)

This expression follows from integration of eq 12 with Cn−1 from eq 6. We note also that an approximation to Nn(t) from eq 14 is eq S31 in SI, which is applicable when kZ/b ≪ 1, for then the kZ/b terms in eq 14 are negligible. The solid lines 2 and 4 in Figure 6a illustrate the Nn(t) dependences from eqs 13 and 14, respectively, at n = 4, a = 0.01 s−1, b = 1 s−1, k = 104 M−1 s−1, and Z = 10 μM (see also Figure S3 in SI). The dotted lines visualize Nn(t) from eq 17 at n = 2 or 4. The corresponding asymptotic Nn(t) dependences from

dNn(t ) = kZCn − 1(t ) (12) dt which results from summation of eq 2 from n to infinity and taking into account that the concentration of infinitely long fibrils is nil throughout fibrillation experiments. Comparison of eqs 7 and 12 shows that, as it should be, dNn/dt = Jn, where Jn is the formation rate of n-sized fibrils. Substitution of C1 and Cn−1 from eqs 3 and 5 in eq 12 and integration under the initial condition Nn(0) = 0 (see SI) yields the exact N2(t) and Nn(t) dependences (n = 3, 4, 5, ...)

Figure 6. Time dependence of (a) the number and (b) the mass concentrations of fibrils of size n ≥ 2 (lines 2) and n ≥ 4 (lines 4): solid lines 2 and 4 in panel a, eqs 13 and 14, respectively; solid lines 2 and 4 in panel b, eqs 22 and 23, respectively; dashed lines in panels a and b, eqs 16 and 24, respectively; dotted lines in panels a and b, eqs 17 and 26, respectively. E


Article


long-time and the short-time limits. Indeed, from eqs 16−18 it follows that (n = 2, 3, 4, ...)

eq 16 are represented by dashed lines (dashed line 2 is not visible on the time scale of the figure). As indicated by arrows, the time intercepts of the asymptotic straight lines represent the delay times θ2 and θ4 from eq 15. Importantly, regardless of the fibril size n, the slopes of these lines are identical and directly give the stationary nucleation rate Jst. For that reason, in experiments on nucleation kinetics, Nn(t) curves like those in Figure 6a are widely used for a highly reliable determination of Jst and θn. Figure 6a evidences that θ4 can be much greater than θ2 (which is equal to the nonstationary period τ of nucleation). This implies that, compared with τ, θn can have a much greater contribution to the fibrillation lag time tlag when this is recorded experimentally by a technique that is not powerful enough to detect even the fibril dimers. This is understandable upon realizing that according to eq 15 θn is merely the sum of τ and the time tgr,n = (n − 2)/Re of growth of a dimer into an n-sized fibril. As seen from eqs 10 and 15, like τ, θn is independent of the activation frequency a and diminishes monotonically with increasing the P-monomer concentration Z. The θn(Z) dependence from eq 15 at n = 4 and k = 104 M−1 s−1 is displayed in Figure 5b, the b values used being 10−4, 0.1, 1, and 5 s−1 (as indicated). Comparing Figures 5a,b, we notice that at small Z values the delay time θ4 of tetramer formation can be more than one order of magnitude longer than the nonstationary nucleation period τ, which is also the delay time θ2 of dimer formation. Thus under otherwise equal conditions experimental techniques with different fibril length detection limits are bound to yield different values (and Z dependences) of the delay time θn of fibril formation. Probability to Form at Least One Fibril. Once the number concentration Nn(t) of fibrils of size n or larger is known, the probability Pn(t) to form at least one of these fibrils until time t is readily obtained from the general and exact formula (see, e.g., ref 3) (n = 2, 3, 4, ...) Pn(t ) = 1 − e−Nn(t )V

Pn(t ) = 1 − exp[−Jst V (t − θn)]

when t ≫ θn and that (n = 2, 3, 4, ...) ⎛ akn − 1Z nV n⎞ Pn(t ) = 1 − exp⎜ − t ⎟ n! ⎠ ⎝ ⎛ J R n − 2V ⎞ e t n⎟⎟ = 1 − exp⎜⎜ − st n ! τ ⎝ ⎠

(21)

when t ≪ θn. We note as well that if the P-monomer concentration Z is so low that kZ/b ≪ 1, then Pn(t) can be calculated from the approximate eq S37 in SI. The solid lines 2 and 4 in Figure 7 graph the Pn(t) dependence at n = 2 and 4, respectively. They are drawn

Figure 7. Time dependence of the probability to form at least one fibril of size n ≥ 2 (lines 2) or n ≥ 4 (lines 4) in solution with volume of (a) 1 and (b) 0.1 μm3: solid lines 2 and 4, eq 19 and eq S36 in SI, respectively; dashed lines, eq 20; dotted lines, eq 21.

(18)

according to eq 19 and eq S36 in SI with a = 0.01 s−1, b = 1 s−1, k = 104 M−1 s−1, and Z = 10 μM for two solution volumes: V = 1 μm3 (Figure 7a) and V = 0.1 μm3 (Figure 7b). The sigmoidal shape of the Pn(t) function is due to the delayed appearance of fibrils of the specified size n. As the fibril delay time θn is longer for the tetramers than for the dimers (θ4 = 0.35 min, θ2 = 0.015 min), P4 lags behind P2. The dashed and dotted lines 2 and 4 illustrate the long-time and short-time Pn(t) dependences from eqs 20 and 21, respectively. Comparing lines 2 (or 4) in Figure 7a,b, we observe that the smaller the solution volume, the better the accuracy of eq 20. Conversely, eq 21 performs better for the larger volume solution. The reason for this effect of the solution volume on Pn is that while for smaller V the change of Pn between the experimentally interesting values of 0.05 and 0.95 may occur when t ≫ θn; for larger V this change could take place when t ≪ θn. Importantly, for a given n, lines 4 in Figure 7 evidence that for some values of V neither eq 20 nor eq 21 is able to describe adequately the Pn(t) dependence. The increase in the time interval of 0.05 to 0.95 change of Pn with decreasing V is also worth noting . This implies greater scatter of the time of appearance of the first fibril of size n or larger in solutions with smaller volumes. Fibril Mass Concentration. The fibril mass concentration Mn is defined as Mn(t) = ∑∞ j=njCj(t) and is another important experimental observable. Because M2 represents the mass concentration of all fibrils in the solution, usually it is this

which is eq S34 in SI and which is valid when the fibrils in the solution with volume V appear randomly and independently of each other. This probability is another experimentally accessible quantity because if a total of Xtot successive or simultaneous experiments are performed under the same conditions and at least one fibril of size n or larger is detected in Xdet of the experiments at a specified time t, Pn at that time is merely equal to the Xdet/Xtot ratio. In fact, the experimental determination of Pn(t) is a simple and precise method for obtaining Nn(t) data from eq 18 presented as Nn(t) = −(1/V) ln[1 − Pn(t)] but solely under conditions at which NnV ≤ 3, for only then is Pn less than or equal to 0.95 and hence reliably measurable. For that reason, small-volume solutions such as those used in microfluidics are particularly suitable for experimental Pn(t) studies. With the help of N2 from eq 13, we find from eq 18 that within the three-parameter CFW model of fibrillation the probability P2 to form at least one fibril of size n ≥ 2 until time t is given by the exact expression P2(t ) = 1 − exp{−Jst V [t − τ(1 − e−t / τ )]}

(20)

(19)

Similarly, substitution of Nn from eq 14 into eq 18 yields the probability Pn at n = 3, 4, 5, ... (as an example, given by eq S36 in SI is P4). The probability Pn has a simple form in both the F


Article


Also, eqs S49 and S50 in SI are approximate Mn(t) equations which are applicable when kZ/b ≪ 1, because then the kZ/b terms in eqs 22 and 23 are negligible. In Figure 6b, the solid lines 2 and 4 visualize the Mn(t) dependences from eq 22 and from eq 23 at n = 4, respectively, and the dashed and dotted lines illustrate eqs 24 and 26. (Dashed line 2 is not visible on the time scale of the Figure.) The lines are drawn with a = 0.01 s−1, b = 1 s−1, k = 104 M−1 s−1, and Z = 10 μM. We see that when the smallest detectable fibrils are the tetramers, the lag in the rise of the fibril mass with time is considerably longer then the lag that would be observed if the dimers could also be detected. This is so because this lag is closely related to the delay time θn = τ + tgr,n (indicated by the arrows in Figure 6b) to which the contribution of the time tgr,n = (n − 2)/Re of dimer growth to the detectable size n can be substantial when n is large enough. The implication concerning the lag time tlag obtained from Mn(t) data is thus that this time can be affected considerably by the fibril length detection limit of the experimental technique used. Mean Fibril Size. As the mean size nmean,n of the fibrils of size n or larger is defined by (n = 2, 3, 4, ...)

quantity that is considered in theoretical studies. However, not every experimental technique is able to detect all fibrils down to the dimers. In experiments, the smallest detectable fibril size may often be an integer substantially greater than 2. Similar to Nn, it is thus important to know Mn for any n ≥ 2. One way to find the Mn(t) dependence corresponding to the three-parameter CFW model is to use Cn from eq 5 in the above definition for Mn. A more convenient way is to first multiply eq 2 by n and then sum the resulting equation from n to infinity. Doing that and performing the necessary integrations (see SI) leads to the exact expressions (n = 3, 4, 5, ...) M 2(t ) = Jst

{

kZ 2 t + (2 − kZτ )[t − τ(1 − e−t / τ )] 2

} (22)

⎡ kZ (n − 2)(n + 1) ⎤ Mn(t ) = Jst ⎢ t 2 + (2 − kZτ )(t − τ ) − ⎥ ⎣ 2 ⎦ 2kZ ⎛ kZ ⎞n − 2 + Jst τ(n − kZτ )⎜ − ⎟ e−t / τ ⎝ b ⎠ −kZt n − 3 ⎧⎡ ⎛ ⎞n − j − 2 ⎤ e + Jst ∑ ⎨⎢1 − ⎜⎝− kZ ⎟⎠ ⎥ kZ j = 0 ⎩⎣ b ⎦

nmean, n(t ) =

⎪

⎪

j

×

∑ i=0

⎫ n−j+i−1 (kZt )i ⎬ i! ⎭

(23)

which are eqs S41 and S42 in SI and in which Jst and τ are given by eqs 9 and 10. In the long-time and short-time limits, the above Mn(t) dependences simplify considerably. When t ≫ θn, the terms with the exponential factors are negligible and eqs 22 and 23 take the form (n = 2, 3, 4, ...) ⎡R (n − 2)(n + 1) ⎤ Mn(t ) = Jst ⎢ e t 2 + (2 − R eτ )(t − τ ) − ⎥ 2R e ⎦ ⎣2 (24)

which reveals that long enough after the initial delay, regardless of the fibril size n, Mn is an increasing quadratic function of t. The t2 term is the leading one and has the classical OK form1,63 (1/2)JstRet2, because within the three-parameter CFW model the fibril elongation rate Re is given by R e = kZ

(25)

Equation 24 thus shows that, like Nn from eq 16, in the longtime limit Mn is governed by the three macroscopic parameters Jst, τ, and Re, which, in turn, are related to the three microscopic parameters a, b, and k via the exact eqs 9, 10, and 25. It is also worth noting that the t2 term in eq 24 is known from the Kolmogorov−Johnson−Mehl−Avrami description of the crystal mass in overall crystallization by nucleation and growth of needle-like crystallites.3,64 Early at the onset of fibrillation, that is, when t ≪ θn, provided the monomer passivation frequency b is finite, Mn is independent of this frequency and is a parabolic function of time given by eq S40 in SI (n = 2, 3, 4, ...) Mn(t ) =

J R en − 2 n akn − 1Z n n t = st t (n − 1)! (n − 1) ! τ

(27)

it is obtainable by substitution of the above Nn(t) and Mn(t) expressions into this equation. For instance, from eqs 16, 24, and 27, it follows that regardless of n, when t ≫ θn, nmean,n increases linearly at half the elongation rate Re = kZ of the separate fibrils: nmean,n(t) ∝ (1/2)Ret. The factor 1/2 takes into account that only the large-size branch of the fibril size distribution Cn moves toward greater n as time goes on (see Figure 4). Initially, however, nmean,n is practically timeindependent. Indeed, for t ≪ θn, from eqs 17, 26, and 27 we have nmean,n(t) ≈ n. Comparison with Experiment. Most of the t and Z dependences of the various quantities presented above are liable to experimental verification. Concerning the nucleation process, Nn(t) and/or Pn(t) data are highly preferable because their analysis is unambiguous in finding Jst and θn (or τ) and unveiling the dependence of these quantities on the monomer concentration or other experimental controllables. However, while such data abound for nucleation of protein and other crystals (see, e.g., refs 3 and 65−78), they are scarce for protein fibrillation (e.g., ref 79) and their obtainment is yet to become a routine. Instead, Mn(t) data are those that are widely reported for fibrillation of proteins. Their analysis, however, is usually equivocal in that “multiple models provide equivalent fits, making mechanistic determination impossible.”17 Additional difficulty with the Mn(t) data is that the size of the smallest detectable fibrils is not always known. For instance, when the thioflavin-T fluorescence method is employed for monitoring the Mn(t) dependence, in analyzing the resulting dependence, it is commonly assumed that even the fibril dimers are detectable (then n = 2) despite that the detectability limit may be at larger fibril size. To confront theory with experiment, we shall use the experimental Mn(t) data of Ellisdon et al.41 for in vitro aggregation of ataxin-3 without a polyglutamine tract (at3(QHQ)), of nonpathological length ataxin-3 with 15 glutamines (at3(Q15)), and of pathological length ataxin-3 with 64 glutamines (at3(Q64)). Ataxin-3 is an intracellular protein that causes the neurodegenerative disorder spinocer-

⎪ ⎪

Mn(t ) Nn(t )

(26) G


Article

The Journal of Physical Chemistry B ebellar ataxia type-3 when its polyglutamine tract is expanded beyond 45 consecutive residues. Ellisdon et al.41 determined the normalized thioflavin-T fluorescence intensity α as a function of time t and, applying Ferrone’s analysis of nucleation-mediated fibrillation,63 calculated nucleus size of 1.15, 0.86, and 0.72 for at3(QHQ), at3(Q15), and at3(Q64), respectively. When rounded to unity, this numbers are physically meaningful if the nucleation process is conceived of as transition from native-state monomer to structurally (and/ or conformationally) changed monomer,41 with the former and the latter being the P-monomer and the A-monomer in Figure 1, respectively. When not rounded to unity, however, the fractional nucleus size is a problem that has recently been addressed by Vitalis and Pappu.21 As we shall see below, analysis of Ellisdon et al.’s41 α(t) data by means of the M2(t) dependence from eq 22 eliminates this problem. In analyzing the data, we posit that (i) initially, virtually all protein monomers in the solution are passive, (ii) the fibrils are practically insoluble, and (iii) α(t) = M2(t)/M2,max, where M2,max is the maximum value of the fibril mass concentration. In addition, we assume (but a posteriori verify) that the condition C1 ≪ Z is satisfied during the early stage of the fibrillation process. In α-vs-t coordinates, the up-triangles, down-triangles, and circles in Figures 8 and 9 represent Ellisdon et al.’s41

Figure 9. Time dependence of the fibrillation degree of three ataxin-3 variants at initial monomer concentration of (a) 60 μM and (b) 80 μM: up-triangles, down-triangles, and circles, experimental data of Ellisdon et al.41 for at3(QHQ), at3(Q15), and at3(Q64), respectively; dashed lines QHQ, Q15, and Q64, best fit of eq 29; solid lines QHQ, Q15, and Q64, eq 28 with the Jst/M2,max, Re, and τ values obtained from the best fit of eq 29.

α(t) < 0.1Z/M2,max restriction is satisfied by the presented data41 at all monomer concentrations and for all three ataxin-3 variants, which is why they are analyzed below. It is mainly the lack of sufficient number of α(t) data complying with this restriction that does not allow using other available experimental results34,38 appropriate for verification of eq 22. To employ eq 22 for analysis of the α(t) data in Figures 8 and 9, we represented it in the form Jst Re 2 α (t ) = t + (2 − R eτ )[t − τ(1 − e−t / τ )] M 2,max 2

{

}

(28)

However, it turned out that this equation could not be fitted to all α(t) data sets in these Figures, presumably because of the scatter (albeit rather small) of the experimental points and/or their insufficient number in the earliest stage (during the first 1 or 2 h) of fibrillation. For that reason, we had to use either the asymptotic α(t) dependence Jst ⎡ R e 2 ⎤ α (t ) = ⎢ t + (2 − R eτ )(t − τ )⎦⎥ ⎣ M 2,max 2 (29)

Figure 8. Time dependence of the fibrillation degree of three ataxin-3 variants at initial monomer concentration of (a) 30 and (b) 40 μM: up-triangles, down-triangles, and circles, experimental data of Ellisdon et al.41 for at3(QHQ), at3(Q15), and at3(Q64), respectively; dashed lines QHQ, Q15, and Q64, best fit of eq 29; solid lines QHQ, Q15, and Q64, eq 28 with the Jst/M2,max, Re, and τ values obtained from the best fit of eq 29.

or the initial α(t) dependence α (t ) =

at3(QHQ), at3(Q15), and at3(Q64) data, respectively, obtained at the onset of fibrillation at initial monomer concentration Z = 30, 40, 60, or 80 μM (as indicated). The data were obtained at fixed total protein concentration so that during the fibrillation process the monomers in the solution were gradually exhausted. As eq 22 is valid at time-independent Z, employing it for analysis of these and similarly obtained M2(t) data is justified only for those of them that are collected in the early stage of fibrillation when the drop in the monomer concentration is such that M2(t) < 0.1Z; besides, the smaller the numerical factor, the more accurate the analysis. In terms of α this restriction reads α(t) < 0.1Z/M2,max. Noting that all Ellisdon et al.’s α(t) data41 correspond to the M2,max value at Z = 80 μM, we can set M2,max = 80 μM because for practically insoluble fibrils M2,max = Z is an acceptable approximation for M2,max. As seen in Figures 8 and 9, with M2,max = 80 μM, the

⎛ 2 2 − R eτ 3⎞ ⎜t − t ⎟ ⎠ M 2,max τ ⎝ 6τ Jst

(30)

While eq 29 approximates eq 28 for t > 3τ (because then the exponential term in eq 28 is negligible) and corresponds to eq 24 at n = 2, eq 30 is valid for t ≪ τ and corresponds to eq 26 at n = 2. (It follows from eq 28 upon expanding the exponential up to the t3 term.) Because of these two limiting α(t) dependences only the first one proved able to provide physically acceptable best fit to the experimental data (see SI), presented below are the results obtained with the aid of eq 29. The dashed lines QHQ, Q15, and Q64 in Figures 8 and 9 illustrate the best fit of eq 29 to the experimental data by means of the three macroscopic free parameters Re, τ, and Jst/M2,max. The lines are drawn with the Re, τ, and Jst/M2,max values obtained from the fit and listed in Table S1 in SI. Listed in this Table as well are the Jst values resulting from recalling that in H


Article

The Journal of Physical Chemistry B the Jst/M2,max ratio, approximately, M2,max = 80 μM. The solid lines QHQ, Q15, and Q64 in Figures 8 and 9 visualize the full α(t) dependence from eq 28 with the Re, τ, and Jst/M2,max values from Table S1 in the SI. Despite the fact that eq 29 (or eq 28) describes quite well the experimental data, the compatibility of the obtained Re, τ, and Jst values with the CFW model in Figure 1a requires that these values satisfy three conditions. The first one is the condition Reτ ≤ 1 that follows from eqs 10 and 25. Table S1 in SI shows that with two exceptions within the statistical error, the Reτ product obeys this condition. Second, the model is applicable when at all times C1 ≪ Z. As many of the data points in Figures 8 and 9 fall in the period when the fibril nucleation is practically stationary (then t > 3τ), this condition takes the form Cst = Jst/ Re ≪ Z, because C1 cannot surpass its stationary value Cst. The last column of Table S1 in SI lists the Cst values resulting from the division of the Jst and Re values in the Table. It is seen that, as required, for all three ataxin-3 variants Cst is much smaller than Z ( 2, the linear Nn(t) dependence from eq 16 may be observed with a considerable delay quantified by θn. For protein solutions with such small volumes V that NnV ≤ 3, it may be experimentally advantageous to investigate not the Nn(t) dependence itself but the related time dependence of the probability Pn to form at least one fibril of size n or larger. When the dimers are the smallest detectable fibrils, the probability P2 to form at least one (n ≥ 2)-sized fibril until time t is expressed by eq 19. Obtaining Nn(t) and/or Pn(t) data is highly desirable because their analysis provides the most reliable information about the fibril stationary nucleation rate Jst and, possibly, the delay time θn. The mass concentration Mn of the fibrils of n or more monomers is given by eqs 22 and 23. Like Nn, Mn is initially bindependent and proportional to tn, as specified by eq 26, but for long enough times, regardless of the fibril size, it depends quadratically on t according to eq 24. The delay time θn controls the initial delay in fibrillation and, especially when the smallest detectable fibrils are considerably longer than the dimers, it can contribute significantly to the fibrillation lag time often reported in experiments at fixed total concentration of protein. The three-parameter CFW model describes self-consistently the early stage of the fibrillation of three ataxin-3 variants, one of them pathological. Analysis of the time course of the fibrillation degree yields the Re(Z), τ(Z), and Jst(Z) dependences and provides values for the frequencies a, b, and k of monomer activation, passivation, and attachment to fibrils. The mechanistic insight gained is that the elongation and stationary nucleation rates Re and Jst of the pathological ataxin-3 variant are higher than those of the nonpathological one because the Pmonomers of the pathological variant transit more frequently into fibrillation-active state, spend longer time in this state, and attach themselves more often to available fibrils.

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b09302. Abbreviations. 1. Fibril size distribution. 2. Fibril formation rate. 3. Fibril number concentration. 4. Probability to form at least one fibril. 5. Fibril mass concentration. 6. Rate equations of the unified model. 7. Comparison with experiment. References. (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Dimo Kashchiev: 0000-0003-3952-3004 Notes

The author declares no competing financial interest.

■

ACKNOWLEDGMENTS Fruitful discussions with Dr. Stefan Auer and his comments concerning the manuscript of the present paper are gratefully acknowledged.

■

REFERENCES

(1) Oosawa, F.; Kasai, M. A theory of linear and helical aggregations of macromolecules. J. Mol. Biol. 1962, 4, 10−21. (2) Abraham, F. F. Homogeneous Nucleation Theory; Academic: New York, 1974. (3) Kashchiev, D. Nucleation: Basic Theory with Applications; Butterworth-Heinemann: Oxford, U.K., 2000. (4) Wegner, A.; Engel, J. Kinetics of the cooperative association of actin to actin filaments. Biophys. Chem. 1975, 3, 215−225. (5) 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. (6) Goldstein, R. F.; Stryer, L. Cooperative polymerization reactions: analytical approximations, numerical examples, and experimental strategy. Biophys. J. 1986, 50, 583−599. (7) Flyvbjerg, H.; Jobs, E.; Leibler, S. Kinetics of self-assembling microtubules: an “inverse problem” in biochemistry. Proc. Natl. Acad. Sci. U. S. A. 1996, 93, 5975−5979. (8) Masel, J.; Jansen, V. A. A.; Nowak, M. A. Quantifying the kinetic parameters of prion replication. Biophys. Chem. 1999, 77, 139−152. (9) Poschel, T.; Brilliantov, N. V.; Frommel, C. Kinetics of prion growth. Biophys. J. 2003, 85, 3460−3474. (10) Roberts, C. J. Kinetics of irreversible protein aggregation: analysis of extended Lumry-Eyring models and implications for predicting protein shelf life. J. Phys. Chem. B 2003, 107, 1194−1207. (11) 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, e321. (12) Hall, D.; Edskes, H. Silent prions lying in wait: A two-hit model of prion/amyloid formation and infection. J. Mol. Biol. 2004, 336, 775−786. (13) Kunes, K. C.; Cox, D. L.; Singh, R. R. P. One-dimensional model of yeast prion aggregation. Phys. Rev. E 2005, 72, 051915. (14) Powers, E. T.; Powers, D. L. The kinetics of nucleated polymerizations at high concentrations: amyloid fibril formation near and above the “supercritical concentration. Biophys. J. 2006, 91, 122− 132. (15) Andrews, J. M.; Roberts, C. J. A Lumry-Eyring nucleated polymerization model of protein aggregation kinetics: 1. Aggregation with pre-equilibrated unfolding. J. Phys. Chem. B 2007, 111, 7897− 7913. J


Article

The Journal of Physical Chemistry B (16) 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. (17) Bernacki, J. P.; Murphy, R. M. Model discrimination and mechanistic interpretation of kinetic data in protein aggregation studies. Biophys. J. 2009, 96, 2871−2887. (18) Knowles, T. P. J.; Waudby, C. A.; Devlin, G. L.; Cohen, S. I. A.; 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. (19) Powers, E. T.; Ferrone, F. A. Kinetic Models for Protein Misfolding and Association. In Protein Misfolding Diseases: Current and Emerging Principles and Therapies; Ramirez-Alvarado, M., Kelly, J. W., Dobson, C. M., Eds.; Wiley: Hoboken, NJ, 2010; pp 73−92. (20) Cohen, S. I. A.; Vendruscolo, M.; Welland, M. E.; Dobson, C. M.; Terentjev, E. M.; Knowles, T. P. J. Nucleated polymerization with secondary pathways. I. Time evolution of principal moments. J. Chem. Phys. 2011, 135, 065105. (21) Vitalis, A.; Pappu, R. V. Assessing the contribution of heterogeneous distributions of oligomers to aggregation mechanisms of polyglutamine peptides. Biophys. Chem. 2011, 159, 14−23. (22) Prigent, S.; Ballesta, A.; Charles, F.; Lenuzza, N.; Gabriel, P.; Tine, L. M.; Rezaei, H.; Doumic, M. An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation. PLoS One 2012, 7, e43273. (23) Hong, L.; Yong, W.-A. Simple moment-closure model for the self-assembly of breakable amyloid filaments. Biophys. J. 2013, 104, 533−540. (24) Schreck, J. S.; Yuan, J.-M. A kinetic study of amyloid formation: fibril growth and length distributions. J. Phys. Chem. B 2013, 117, 6574−6583. (25) Michaels, T. C. T.; Garcia, G. A.; Knowles, T. P. J. Asymptotic solutions of the Oosawa model for the length distribution of biofilaments. J. Chem. Phys. 2014, 140, 194906. (26) Garcia, G. A.; Cohen, S. I. A.; Dobson, C. M.; Knowles, T. P. J. Nucleation-conversion-polymerization reactions of biological macromolecules with prenucleation clusters. Phys. Rev. E 2014, 89, 032712. (27) Michel, D. Modeling generic aspects of ideal fibril formation. J. Chem. Phys. 2016, 144, 035101. (28) Murray, B.; Sorci, M.; Rosenthal, J.; Lippens, J.; Isaacson, D.; Das, P.; Fabris, D.; Li, S.; Belfort, G. A2T and A2V Aβ peptides exhibit different aggregation kinetics, primary nucleation, morphology, structure, and LTP inhibition. Proteins: Struct., Funct., Genet. 2016, 84, 488−500. (29) Morris, A. M.; Watzky, M. A.; Finke, R. G. Protein aggregation kinetics, mechanism, and curve-fitting: a review of the literature. Biochim. Biophys. Acta, Proteins Proteomics 2009, 1794, 375−397. (30) Gillam, J. E.; MacPhee, C. E. Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth. J. Phys.: Condens. Matter 2013, 25, 373101. (31) Campioni, S.; Monsellier, E.; Chiti, F. Why Proteins Misfold? In Protein Misfolding Diseases: Current and Emerging Principles and Therapies; Ramirez-Alvarado, M., Kelly, J. W., Dobson, C. M., Eds.; Wiley: Hoboken, NJ, 2010; pp 3−20. (32) Griffith, J. S. Self-replication and scrapie. Nature 1967, 215, 1043−1044. (33) Jarrett, J. T.; Lansbury, P. T., Jr. Seeding “one-dimensional crystallization” of amyloid: a pathogenic mechanism in Alzheimer’s disease and scrapie? Cell 1993, 73, 1055−1058. (34) Chen, S.; Ferrone, F. A.; Wetzel, R. Huntington’s disease age-ofonset linked to polyglutamine aggregation nucleation. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 11884−11889. (35) Marchal, S.; Shehi, E.; Harricane, M.-C.; Fusi, P.; Heitz, F.; Tortora, P.; Lange, R. Structural instability and fibrillar aggregation of non-expanded human ataxin-3 revealed under high pressure and temperature. J. Biol. Chem. 2003, 278, 31554−31563.

(36) Hurshman, A. R.; White, J. T.; Powers, E. T.; Kelly, J. W. Transthyretin aggregation under partially denaturing conditions is a downhill polymerization. Biochemistry 2004, 43, 7365−7381. (37) Ahmad, A.; Millett, I. S.; Doniach, S.; Uversky, V. N.; Fink, A. L. Stimulation of insulin fibrillation by urea-induced intermediates. J. Biol. Chem. 2004, 279, 14999−15013. (38) Ignatova, Z.; Gierasch, L. M. Aggregation of a slow-folding mutant of a β-clam protein proceeds through a monomeric nucleus. Biochemistry 2005, 44, 7266−7274. (39) Wetzel, R. Kinetics and thermodynamics of amyloid fibril assembly. Acc. Chem. Res. 2006, 39, 671−679. (40) Ellisdon, A. M.; Thomas, B.; Bottomley, S. P. The two-stage pathway of ataxin-3 fibrillogenesis involves a polyglutamineindependent step. J. Biol. Chem. 2006, 281, 16888−16896. (41) Ellisdon, A. M.; Pearce, M. C.; Bottomley, S. P. Mechanisms of ataxin-3 misfolding and fibril formation: kinetic analysis of a diseaseassociated polyglutamine protein. J. Mol. Biol. 2007, 368, 595−605. (42) Nam, H. B.; Kouza, M.; Zung, H.; Li, M. S. Relationship between population of the fibril-prone conformation in the monomeric state and oligomer formation times of peptides: insights from all-atom simulations. J. Chem. Phys. 2010, 132, 165104. (43) Kar, K.; Jayaraman, M.; Sahoo, B.; Kodali, R.; Wetzel, R. Critical nucleus size for disease-related polyglutamine aggregation is repeatlength dependent. Nat. Struct. Mol. Biol. 2011, 18, 328−336. (44) Kar, K.; Hoop, C. L.; Drombosky, K. W.; Baker, M. A.; Kodali, R.; Arduini, I.; van der Wel, P. C. A.; Horne, W. S.; Wetzel, R. βhairpin-mediated nucleation of polyglutamine amyloid formation. J. Mol. Biol. 2013, 425, 1183−1197. (45) Ferrone, F. A. Assembly of Aβ proceeds via monomeric nuclei. J. Mol. Biol. 2015, 427, 287−290. (46) Ranganathan, S.; Ghosh, D.; Maji, S. K.; Padinhateeri, R. A minimal conformational switching-dependent model for amyloid selfassembly. Sci. Rep. 2016, 6, 21103. (47) Pellarin, R.; Caflisch, A. Interpreting the aggregation kinetics of amyloid peptides. J. Mol. Biol. 2006, 360, 882−892. (48) Bellesia, G.; Shea, J.-E. Effect of β-sheet propensity on peptide aggregation. J. Chem. Phys. 2009, 130, 145103. (49) Bellesia, G.; Shea, J.-E. Diversity of kinetic pathways in amyloid fibril formation. J. Chem. Phys. 2009, 131, 111102. (50) Bieler, N. S.; Knowles, T. P. J.; Frenkel, D.; Vacha, R. Connecting macroscopic observables and microscopic assembly events in amyloid formation using coarse grained simulations. PLoS Comput. Biol. 2012, 8, e1002692. (51) Saric, A.; Chebaro, Y. C.; Knowles, T. P. J.; Frenkel, D. Crucial role of nonspecific interactions in amyloid nucleation. Proc. Natl. Acad. Sci. U. S. A. 2014, 111, 17869−17874. (52) Saric, A.; Michaels, T. C. T.; Zaccone, A.; Knowles, T. P. J.; Frenkel, D. Kinetics of spontaneous filament nucleation via oligomers: insights from theory and simulation. J. Chem. Phys. 2016, 145, 211926. (53) Haezebrouck, P.; Joniau, M.; van Dael, H.; Hooke, S. D.; Woodruff, N. D.; Dobson, C. M. An equilibrium partially folded state of human lysozyme at low pH. J. Mol. Biol. 1995, 246, 382−387. (54) Auer, S.; Miller, M. A.; Krivov, S. V.; Dobson, C. M.; Karplus, M.; Vendruscolo, M. Importance of metastable states in the free energy landscapes of polypeptide chains. Phys. Rev. Lett. 2007, 99, 178104. (55) Tobacman, L. S.; Korn, E. D. The kinetics of actin nucleation and polymerization. J. Biol. Chem. 1983, 258, 3207−3214. (56) Gamez, P.; Caballero, A. B. Copper in Alzheimer’s disease: implications in amyloid aggregation and neurotoxicity. AIP Adv. 2015, 5, 092503. (57) Linse, S.; Cabaleiro-Lago, C.; Xue, W.-F.; Lynch, I.; Lindman, S.; Thulin, E.; Radford, S. E.; Dawson, K. A. Nucleation of protein fibrillation by nanoparticles. Proc. Natl. Acad. Sci. U. S. A. 2007, 104, 8691−8696. (58) Bellucci, L.; Ardevol, A.; Parrinello, M.; Lutz, H.; Lu, H.; Weidner, T.; Corni, S. The interaction with gold suppresses fiber-like conformations of the Amyloid β (16−22) peptide. Nanoscale 2016, 8, 8737−8748. K


Article

The Journal of Physical Chemistry B (59) Powers, E. T.; Powers, D. L. Mechanisms of protein fibril formation: nucleated polymerization with competing off-pathway aggregation. Biophys. J. 2008, 94, 379−391. (60) Kashchiev, D.; Auer, S. Nucleation of amyloid fibrils. J. Chem. Phys. 2010, 132, 215101. (61) Kashchiev, D.; Cabriolu, R.; Auer, S. Confounding the paradigm: peculiarities of amyloid fibril nucleation. J. Am. Chem. Soc. 2013, 135, 1531−1539. (62) Kashchiev, D. Protein polymerization into fibrils from the viewpoint of nucleation theory. Biophys. J. 2015, 109, 2126−2136. (63) Ferrone, F. Analysis of protein aggregation kinetics. Methods Enzymol. 1999, 309, 256−274. (64) Auer, S.; Kashchiev, D. Insight into the correlation between lag time and aggregation rate in the kinetics of protein aggregation. Proteins: Struct., Funct., Genet. 2010, 78, 2412−2416. (65) Turnbull, D.; Cormia, R. L. Kinetics of crystal nucleation in some normal alkane liquids. J. Chem. Phys. 1961, 34, 820−831. (66) Gutzow, I. Induced crystallization of glass-forming systems: a case of transient heterogeneous nucleation (Part II). Contemp. Phys. 1980, 21, 243−263. (67) Fokin, V. M.; Kalinina, A. M.; Filipovich, V. N. Nucleation in silicate glasses and effect of preliminary heat treatment on it. J. Cryst. Growth 1981, 52, 115−121. (68) Galkin, O.; Vekilov, P. G. Are nucleation kinetics of protein crystals similar to those of liquid droplets? J. Am. Chem. Soc. 2000, 122, 156−163. (69) Laval, P.; Salmon, J.-B.; Joanicot, M. A microfluidic device for investigating crystal nucleation kinetics. J. Cryst. Growth 2007, 303, 622−628. (70) Laval, P.; Crombez, A.; Salmon, J.-B. Microfluidic droplet method for nucleation kinetics measurements. Langmuir 2009, 25, 1836−1841. (71) Leng, J.; Salmon, J.-B. Microfluidic crystallization. Lab Chip 2009, 9, 24−34. (72) Selimovic, S.; Jia, Y.; Fraden, S. Measuring the nucleation rate of lysozyme using microfluidics. Cryst. Growth Des. 2009, 9, 1806−1810. (73) Jiang, S.; ter Horst, J. H. Crystal nucleation rates from probability distributions of induction times. Cryst. Growth Des. 2011, 11, 256−261. (74) Ildefonso, M.; Candoni, N.; Veesler, S. Using microfluidics for fast, accurate measurement of lysozyme nucleation kinetics. Cryst. Growth Des. 2011, 11, 1527−1530. (75) Nanev, C. N. Kinetics and intimate mechanism of protein crystal nucleation. Prog. Cryst. Growth Charact. Mater. 2013, 59, 133−169. (76) Dimitrov, I. L.; Hodzhaoglu, F. V.; Koleva, D. P. Probabilistic approach to lysozyme crystal nucleation kinetics. J. Biol. Phys. 2015, 41, 327−338. (77) Brandel, C.; ter Horst, J. H. Measuring induction times and crystal nucleation rates. Faraday Discuss. 2015, 179, 199−214. (78) Hammadi, Z.; Grossier, R.; Zhang, S.; Ikni, A.; Candoni, N.; Morin, R.; Veesler, S. Localizing and inducing primary nucleation. Faraday Discuss. 2015, 179, 489−501. (79) Galkin, O.; Vekilov, P. G. Mechanisms of homogeneous nucleation of polymers of sickle cell anemia hemoglobin in deoxy state. J. Mol. Biol. 2004, 336, 43−59.

L


Modeling the Effect of Monomer Conformational Change on the Early

Recommend Documents