Statistical Mechanics of Globular Oligomer Formation by Protein

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Statistical Mechanics of Globular Oligomer Formation by Protein Molecules Alexander John Dear, Andela Saric, Thomas C.T. Michaels, Christopher M. Dobson, and Tuomas P.J. Knowles J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b07805 • Publication Date (Web): 18 Oct 2018 Downloaded from http://pubs.acs.org on October 20, 2018

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The Journal of Physical Chemistry

Statistical Mechanics of Globular Oligomer Formation by Protein Molecules Alexander J. Dear,† Anđela Šarić,‡,¶ Thomas C. T. Michaels,†,§ Christopher M. Dobson,† and Tuomas P. J. Knowles∗,†,k †Centre for Misfolding Diseases, Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK ‡Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, Gower Street, London WC1E 6BT, UK ¶Corresponding author for simulations. Email: [email protected] §Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA kCavendish Laboratory, Department of Physics, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK E-mail: [email protected]

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Abstract The misfolding and aggregation of proteins into linear fibrils is widespread in human biology, for example in connection with amyloid formation and the pathology of neurodegenerative disorders such as Alzheimer’s and Parkinson’s diseases. The oligomeric species that are formed in the early stages of protein aggregation are of great interest, having been linked with the cellular toxicity associated with these conditions. However, these species are not characterized in any detail experimentally, and their properties are not well understood. Many of these species have been found to have approximately spherical morphology, and to be held together by hydrophobic interactions. We present here an analytical statistical mechanical model of globular oligomer formation from simple idealized amphiphilic protein monomers, and show that this correlates well with Monte-Carlo simulations of oligomer formation. We identify the controlling parameters of the model, which are closely related to simple quantities that may be fitted directly from experiment. We predict that globular oligomers are unlikely to form at equilibrium in many polypeptide systems, but instead form transiently in the early stages of amyloid formation. We contrast the globular model of oligomer formation to a wellestablished model of linear oligomer formation, highlighting how the differing ensemble properties of linear and globular oligomers offer a potential strategy for characterising oligomers from experimental measurements.

Introduction The process of linear self-assembly, whereby monomeric protein molecules aggregate spontaneously to form large filamentous structures, is a central feature of both normal and aberrant human biology. Such a process is, for example, a fundamental feature of the pathology of a range of protein aggregation disorders such as Alzheimer’s disease (AD), Parkinson’s disease (PD), 1–9 sickle-cell anaemia, 10,11 type 2 diabetes 12–14 and the prion diseases. 15–19 Kinetic modelling of these processes has been an active area of research over the past 60 years. The first successful model was developed for actin polymerization and solved analytically 2


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by Oosawa et al in 1962. 20 The model included the formation of new fibrils via a slow, coarse-grained “primary nucleation” process, with monomeric protein molecules associating directly with each other. Once formed, fibrils could grow rapidly by elongation, a universal feature of linear self-assembly. In the 1980s, Eaton & co-workers found that this simple model was insufficient to describe the polymerization of sickle cell haemoglobin, and that an additional, heterogeneous, “secondary nucleation” reaction step was required, 21 in which new fibrils could be formed by the interaction of monomers with the surfaces of existing fibrils. They developed kinetic equations for this new 3-step aggregation process, and solved them for early times. 22–24 Later, it was discovered that prion proliferation could also be described as a linear self-assembly process, and suitable rate equations were devised, featuring primary nucleation and elongation as well as a new “fragmentation” reaction step describing fibril breakage. 25 These rate equations have all been solved analytically in the past decade, 26–30 permitting their application to the interpretation of kinetic data on amyloid aggregation. 31–34 In many of the conditions mentioned above, it has been shown that cellular damage occurs and can be primarily attributed to small pre-fibrillar species, generally described as oligomers, that form early in the aggregation process, 35–37 and also during secondary nucleation, 31,38 rather than the large fibrillar aggregates that are ultimately formed. 39 A key issue is therefore to understand the physicochemical basis for the formation of protein oligomers. Whilst many studies of protein aggregation have focussed on the formation of fibrils, experimental studies of the oligomeric species are relatively less well developed despite their importance in the context of disease. As a result, the diversity and biological and chemical properties of these species are still poorly understood. There are major challenges in detecting oligomers under physiological conditions, where they are typically present at very low concentrations. Recent advances in single molecule detection techniques have, however, greatly enhanced our ability to observe protein oligomers at low concentrations, and to allow estimates to be made of their size distributions. 40,41 In addition, AFM and EM imaging have enabled visualization of protein oligomers, and have re-

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vealed a range of morphologies, including discs, 42 rings, 43,44 chains, 44 and spheres. 44,45 While some of these structures have been determined under physiological conditions, others were reported under artificial conditions; 42 and some may be induced by the methods of preparation and imaging. Several studies have investigated oligomer size distributions, 46–48 although such measurements often involve a range of assumptions and are usually reported as “apparent sizes”. Recently, kinetic models have been developed and applied to experimental data on fibril formation via oligomeric intermediates; 49–52 however, these models treat oligomers in a very coarse-grained way, giving approximate total concentrations of oligomer populations that likely include a range of different oligomer sizes. Statistical mechanical modelling of oligomer formation provides an opportunity to complement and enhance the interpretation of experimental data. Developing these models could have important implications for understanding the kinetics and thermodynamics of protein aggregation, the mechanisms behind oligomer formation, and potentially the origins of their effects on cellular systems. In this paper, we demonstrate that a simple statistical mechanical model for micelles, when appropriately formulated, may be used to model the formation of globular protein oligomers via nonspecific interactions. We investigate the properties of such oligomers using the model and discuss how their key parameters might be determined experimentally. Finally we compare the results of the study to those of linear oligomers using a well-established model, 53,54 and discuss how the different geometries can be distinguished in practice. We validate both linear and globular models using Monte-Carlo simulations. The Supplemental Information (SI) contains additional derivations.

Theoretical Methods Using the theoretical framework offered by the grand canonical ensemble (see SI), a general expression for the concentration f (j) of noninteracting oligomers formed from j monomers in a system at temperature T can be found, in terms of the internal oligomer partition

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function qint (j) = exp(−β∆G◦int (j)) and the fundamental volume v0 (j) associated with the translational partition function:

f (j) =

1 qint (j) eβjµ , NA v0 (j)

(1)

where β = 1/(kB T ) is the inverse temperature, NA is Avogadro’s number, and µ is the chemical potential of the monomers, which is set implicitly by the conservation-of-mass P∞ condition, p = j=1 jf (j), where p is the total monomer concentration. For notational consistency with previous kinetic and thermodynamic studies of oligomerization, 51,55 we hereafter write f (1) = m. We further assume that v0 (j) ≡ v0 is independent of oligomer size. From Eq. (1), it follows that to model equilibrium oligomer size distributions, we must identify the internal free energy change associated with oligomer formation, ∆G◦int (j).

Globular morphology Virtually all proteins have both hydrophobic and hydrophilic sections within their sequence. It is therefore expected that disordered polypeptide chains will typically have some degree of amphiphilic character, or will be able to access an amphiphilic conformation relatively easily. Amphiphiles are able to form spherical micelle-like structures under appropriate conditions through hydrophobic bonding, provided the hydrophobic region is not too large. 56 As a result, we expect that most proteins are capable of forming globular oligomers, and that globular morphology is therefore common. Since hydrophobic bonds are non-directional and can therefore form rapidly, we expect these morphologies to be especially common in the early stages of an aggregation reaction, and that globular oligomers are typically the first aggregate species to form. Such aggregates might be expected to play the role of intermediates en route to the more structured β-sheet-containing species that are ultimately formed. These inferences are in agreement with prior simulations and experimental studies of transient oligomer formation. 2,57–63

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Since globular oligomers formed via hydrophobic interactions are expected to be micellelike structures, we hereafter consider a simplified model of a protein system consisting of an amphiphilic monomeric species that may assemble into small spherical micellar protein oligomers. Our idealized protein monomers have a single hydrophilic region connected to a single hydrophobic region. In addition to developing analytical expressions for the properties of this model, we also carry out Monte-Carlo simulations on a system of this form.

Internal free energy An analytical expression for the free energy required to form a spherical micelle from j monomers has previously been derived by Maibaum, Dinner and Chandler: 64

∆G◦int (j) = α j 2/3 + ∆Gbulk j + h j 5/3 /β,

(2)

where ∆Gbulk is the favourable free energy of transferring the hydrophobic segment of an amphiphile from solution into a hypothetical new oil-like phase consisting solely of hydrophobic segments; and αj 2/3 is an energy penalty arising from surface tension between the bulk water-based phase and the hypothetical hydrophobic phase. hj 5/3 is a connectivity-enforcing term that penalises the separation of hydrophilic and hydrophobic segments that occurs if the micelle becomes too large whilst retaining its spherical geometry, 65 originally computed by utilizing an analogy between micelle formation and charge repulsion. 66 The value of h can usually be related to the geometry of the amphiphile within the micelle, permitting its approximate calculation. 64 The large-j free energy penalty arising from steric clash between hydrophilic segments at the surface of the micelle can be shown to be insignificant in comparison with this term and thus unnecessary for a qualitative description of the system behaviour (see SI). Note that, in the case of charged monomers, the free energy penalty originating from charge repulsion has the same scaling with j as the connectivity-enforcing term, and thus does not require an explicit extra term.

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This mean-field model was developed for typical surfactant micelles with aggregation numbers 50 < j < 100 and whose constituent amphiphiles have long hydrocarbon segments that are hydrophobic, for which the concept of a micelle surface tension is well-defined. For the small aggregation numbers (5 < j < 15) 59 seen in some protein oligomers, however, the hydrophobic groups of the micelle can hardly be considered a distinct phase separated from the bulk phase by a well-defined surface, and it is not clear that this model will remain valid. We instead write the free energy of forming a micelle core as the sum of pairwise interactions between amphiphiles, yielding an overall free energy:

∆G◦int (j) = j

zj Gb + h j 5/3 /β, 2

(3)

where Gb is a pairwise interaction free energy, and zj is a size-dependent average coordination number. For nonspecific interactions we expect this mean-field description to be reasonable; 67 multiple stable structural isomers are not expected to be important except for in clusters of size j = 3 − 4 (e.g. tetrahedra vs squares), which are not expected to contribute significantly to the overall size distribution. This approach for modelling small micellar oligomers was first used in ref. 68. Now zj must have a spherical form such that Eq. (3) recovers Eq. (2) in the limit of large j, and must also reproduce the coordination numbers seen in oligomer ensembles. The simplest way to achieve this is to have a constant term and a j −1/3 term, which is equivalent to assuming that the functional form of Eq. (2) holds even in this size regime. Although the expression employed in ref. 68 has this form, it also has some deficiencies, yielding z3 > 2, and z∞ ' 9.6. Since the maximum coordination number possible in a bulk close-packed structure is well-known from the field of crystallography to be 12, we believe z∞ = 12 is more reasonable, and build this directly into our model, writing:

zj = z∞ (1 − γ j −1/3 ),

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(4)


where z∞ = 12, and γ an as-yet undetermined parameter. Instead of fitting directly to total oligomer concentration data as in ref., 68 we can determine γ by fitting to the maximum possible average coordination numbers for close-packed spherical clusters consisting of approximately isotropic monomers at j = 5 − 8 (trigonal bipyramid, octahedron, etc.) We do not fit at j = 3 − 4 for reasons given above, and we do not fit at j > 8 due to the difficulty of determining ideal geometries at these larger sizes. This approach yields γ = 1.21. Furthermore, we carried out Monte-Carlo simulations of oligomer formation from model amphiphilic monomers (Figure 1a, see below for full details); average coordination numbers measured directly from these simulations agree remarkably well with those predicted by Eq. (4) with this value of γ (Figure 1b). This demonstrates that the model proposed by Maibaum et al (ref. 64) indeed remains valid in this small micelle size regime. Maibaum et al (ref. 64) advised constraining α from Eq. (2) experimentally by directly measuring surface tension from experiments featuring a water phase and a phase consisting of hydrophobic oil molecules. However, this is not readily possible as a representation of amphiphilic proteins rather than hydrocarbon surfactants. We note that such a strategy is often unsuitable even for conventional micelles, with Eq. (2) combined with an experimentally determined α being unable to reproduce Monte Carlo simulations of size distributions 69 consistently. Our formulation of the model therefore offers a distinct advantage for amphiphiles with relatively globular hydrophobic segments by being easier to constrain than Eq. (2). Note that for amphiphiles with elongated hydrophobic segments the form of the coordination number function may need to change, or abandoned altogether in favour of Eq. (2); however, it suffices in the present case and proves the validity of the functional form of Eq. (2) even for the small clusters considered here.

Size distribution Having determined an expression for the free energy of globular oligomerization, we are now in a position to determine the size distribution for globular oligomers. Using (3) in (1), we 8


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a

b Hydrophilic segment Amphiphilic proteins

Hydrophobic segment

Micellar oligomers

Coordination number / z j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


j ~ 5-15

Flexible structure

7 6 5

Obeys spherical scaling

4 3 zj = 12(1-γ/j1/3)

2 1 0

4

8 12 16 Oligomer size / j

Figure 1: a: Schematic of cluster oligomer model. Proteins are modelled simply by a non-bonding hydrophilic region and a hydrophobic region capable of non-directional bonding. Bonding interactions are counted pairwise. Typical protein oligomer aggregation numbers are expected to range from 5-15 monomers. b: We replace bulk and surface energy terms with a pairwise interaction energy term depending on an average spherical coordination number function. This function is in turn determined by fitting a spherical scaling function to the maximum possible average coordination numbers for close-packed spherical clusters at j = 5 − 8 (bars). This matches the results from Monte Carlo simulations (data points), reflecting the flexibility of oligomer structures. Minor deviations giving data points below the function values at larger sizes suggests a tendency to form spherocylindrical oligomers to maintain microphase separation at large j; this effect has been observed previously for Monte Carlo simulations of larger micelles. 70

find: 1 zj 5/3 f (j) = exp −βj Gb − h j + βjµ . NA v0 2

(5)

The origin of this size distribution in micelle theory ensures a total protein transition concentration p∗ , above which the size distribution is peaked, and below which it monotonically decreases. The equilibrium monomer concentration m∗ associated with the transition is analogous to the critical micelle concentration (CMC). A further key advantage of Eq. (5) over the Maibaum-Dinner-Chandler formulation is that we can more naturally interpret the properties of the size distribution in terms of its parameters. We expect the modal oligomer size above p∗ to be controlled by the ratio βGb /h: the trade-off between the interaction free energy, that increases the favourability of larger oligomers, and the connectivity term, that decreases it. Given a known average oligomer size, we then expect the polydispersity of the size distribution to be set by the magnitude of βGb or h; the smaller their magnitude the flatter the internal free energy, and the smaller the difference in stability between oligomers of different sizes. The concentration units are

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then set by v0 . We were able to fit successfully our model globally to our Monte-Carlo simulations of oligomer formation at two different total protein concentrations (Figure 2), yielding βGb /h = −3.62, h = 0.58, and v0 = (11.8 nm)3 . The h value is comparable to that predicted by ab initio calculations 65 (see SI). Details of the simulations are given below. b

25

p = 55 μM

20 15 10 5 0

2

Fraction of oligomers / %

a Fraction of oligomers / %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4 6 8 10 12 14 Oligomer size / j

p = 106 μM

12 10 8 6 4 2 0

2


Figure 2: The analytical size distribution model (bars) can be globally fitted to coarse-grained numerical Monte Carlo results (data points) with reasonable accuracy (fitted parameters in text).

Monte Carlo simulations To study the formation of linear and globular oligomers we used Monte Carlo (MC) simulations, where N = 600 (in case of the linear model) or N = 1000 (in case of the globular model) monomers were placed inside a cubic box to achieve the target concentration. Simulations were run for at least 3 · 108 MC steps. The mapping between the simulation and physical units was done by using the σ = 2nm mapping, where σ is the length-scale in our simulations. This mapping was chosen such that the simulation model reproduces the sizes of typical small proteins, such as Aβ. The size distribution of the resulting oligomers was analysed using an in-house clustering algorithm.

Linear oligomers A monomer is described as a hard sphere of a diameter σ. We place two interaction centres at a distance of 0.4σ from its centre, diametrically opposite from each other. Two monomers 10


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interact with enthalpy − if a pair of interaction centres, one from each monomer, lie less than 0.3σ apart. Such a geometric arrangement creates an effective short-ranged potential by which a single ideal sphere cannot interact with more than one other monomer at a time, giving rise to linear oligomers. A value for of −15kT was used. We found that such a value of the interaction parameter produced a well-sampled size distribution of oligomers at the chosen range of protein concentrations, while keeping the system in a predominantly monomeric state, as has been observed observed in most experimental studies of protein oligomers to date. 50–52,55

Globular oligomers A monomer is described as a hard spherocylinder of a diameter of σ and a length of 4σ. Such a size ratio was initially chosen to mimic the geometry of the Aβ peptide in its betasheet-prone state. 59 An interaction centre is placed at one pole of the spherocylinder, at a distance 1.5σ from its centre. Two spherocylinders interact with the interaction enthalpy of − if their interaction centres are less than 1.3σ apart. This shape of the protein and the related geometric parameters are chosen such that they drive the formation of finite, micellelike oligomers, where tips of participating monomers are in contact in the micelle centre. 59 A value for of −5kT was used. We find that this value of the interaction parameter gives rise to a well-sampled distribution of small oligomers at the chosen range of peptide concentrations. Moreover, at the chosen value of and the peptide concentrations the system is below the critical micelle concentration.

Results & Discussion Key Characteristics of Globular Protein Oligomers at Equilibrium Having predicted how the model parameters will relate to the key properties of the system, we now seek explicit expressions for these properties. An important advantage of doing 11



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so is that these properties are more easily measurable by experiment than is the full size distribution itself; by measuring them we can thus constrain the model and simulate the expected size distribution. Furthermore, we can rewrite Eq. (5) in a particularly convenient form in terms of these properties.

Average oligomer size The most probable oligomer size above m∗ is a characteristic property often directly measurable from experiment, and is given by solving df (j)/dj = 0. A lower bound on its value is given by the location of the point of inflection j ∗ of the size distribution at m = m∗ (see SI; first derived for micelles in Ref. 64):

j∗ = −

6γβGb . 5h

(6)

Maibaum et al. 64 also derived an approximate expression for the most likely micellar size, by approximating the size distribution above the CMC as monodisperse at size j = j 0 . Combining Eq’s (16) and (17) from Ref. 64 using our notation, the value of j 0 is given by solving:

∂β∆G/j ∂j

= j=j 0

1 z∞ γβGb 0−4/3 2h 0−1/3 j + j = 02 ln (fj NA v0 ) , 6 3 j

(7)

where x is the fraction of monomers incorporated into micelles (x = j 0 fj /f1 ). The RHS, small for the large values of j 0 expected for micelles, was further approximated as zero to arrive at an analytical expression for j 0 . We note, however, that the RHS actually approaches zero from below as fj increases towards 1/NA v0 M. We further note that this limit is approximately the highest concentration possible for the system, corresponding to a volume fraction of unity, since v0 is of the order of the monomer dimensions; therefore their expression is in fact an upper bound on the peak position. In particular, from the inequality: z∞ γβGb −4/3 2h −1/3 jp + jp < 0, 6 3 12


(8)

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one obtains the following expression for the peak jp : jp < j 0 = −

6γβGb . 2h

(9)

We expect the peak jp to increase rapidly from j ∗ , as the lower bound j ∗ is an inflection point, and then to approach the upper bound j 0 asymptotically. Therefore, jp will typically be closer to j 0 than to j ∗ . Indeed, in the current case the choice of jp = −5γβGb /2h is in good agreement with our system over a wide range of concentrations (' 9; see Figure 3a-b). Crucially, both bounds show the expected scaling behaviour Gb /h (Figure 3c-d). Only a small increase in p above p∗ is required for most of the monomers in the system to become incorporated into micelles, with a higher peak size, or |Gb |/h, corresponding to a smaller increase required. This expectation is inferred from the fact that above p = p∗ the oligomer concentration scales initially approximately as pjp . Indeed, explicit calculations using parameters fitted from the Monte-Carlo simulations demonstrate that the total protein concentration for which 50% of the total protein molecules, p50 , are in oligomers is only slightly higher than p∗ (Figure 3a, Figure 3c, Figure 3e).

Critical concentration and rescaling Size distributions have units of concentration and, therefore, rescaling by an experimentallymeasurable concentration should nondimensionalize the model, effectively replacing one degree of freedom with another more accessible one. A suitable concentration for globular oligomers is the CMC m∗ , at which a new maximum first appears in the size distribution; this can be shown to be (see SI): 1 exp 6β Gb + (5h)1/3 −6γβ Gb )2/3 NA v0 z ∗ 1 j = exp β Gb , NA v0 2

m∗ =

13


(10a) (10b)


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where zj ∗ is the coordination number at the point of inflection of the size distribution at f (1) = m∗ . Interestingly, the exponent is just the negative of the interaction free energy per monomer within an oligomer of size j ∗ . We can then identify the convenient dimensionless form for the size distribution: z zj ∗ j fˆ(j) = fˆ(1)j exp −βGb j − (j − 1) − hj 5/3 , 2 2

(11)

where fˆ(j) = f (j)/m∗ , and the exponential term is the size distribution at fˆ(1) = 1, which is monotonically decreasing apart from a point of inflection at j ∗ . We clearly see that at concentrations below fˆ(1) = 1, the size distribution is monotonically decreasing, and above m∗ it has a peak. Rescaling concentrations by m∗ thus eliminates v0 and collapses all possible size distributions onto a set of curves indexed only by Gb and h (Figure 3a, Figure 3c, Figure 3e). To map to SI units we now need to determine m∗ instead of v0 , which may often be conveniently measured experimentally. This formulation of the size distribution also allows for a particularly clear interpretation of the effects of varying the parameters. As determined above, the position of the point of inflection is determined by the ratio βGb /h. Increasing −βGb also amplifies the initial decrease in free energy, and thus concentration, for j < j ∗ . Increasing h increases the steepness of the decline of free energy for j > j ∗ . Increasing −βGb and reducing h also have the secondary effect of slightly increasing zj ∗ which slightly increases the initial decline in oligomer concentration with j ∗ . The magnitude of βGb is thus primarily responsible for the magnitude of the increase in p required to transition from a dimer-dominated size distribution to a highly peaked size distribution. Only a moderate −βGb is required for the size distribution to be highly dimer-dominated slightly below fˆ(1) = 1.

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Polydispersity of the size distribution An indicator of the polydispersity of the globular size distribution above m∗ is given by its second derivative with respect to j at j = jp normalized by the peak height, which we label C: 1 d2 f (j) C= . f (j) dj 2 j=jp

(12)

Using Eq. (3) in Eq. (12), we find:

C=

1 1/3

9 jp

γz∞ (−βGb ) − 10 h . jp

(13)

Interestingly there is no explicit concentration dependence; thus j ∗ is always a point of inflection with C = 0, and C is always negative at higher j since the positive first term is reduced more than the negative second term. We see that the polydispersity of the distribution is thus controlled by the magnitude of Gb or h, as expected (Figure 3e-f). Upon closer inspection, doubling (−βGb ) reduces C by 21/3 , whereas halving h reduces it by 24/3 ; thus, the polydispersity is substantially more sensitive to h than to Gb . Halving both |Gb | and h gives no change in jp and thus halves C. Our model considers only spherical oligomers; however, at sizes above jp we expect spherocylindrical oligomers to become important, 70 since it becomes favourable to reduce the average coordination number to avoid significant separation of hydrophilic and hydrophobic protein segments. We therefore expect our model to underestimate the polydispersity. Fitting our model to full size distributions will then only lead to lower bounds on the magnitudes of |Gb | and h, and consequently to upper bounds on v0 , rather than exact values. Indeed, fitting to the Monte-Carlo simulations yielded v0 = (11.8 nm)3 , which is slightly larger than expected for monomer dimensions of (2 nm)3 . However, given only m∗ and j ∗ , the model still has good predictive power apart from the concentration range immediately above p = p∗ . Should greater accuracy for parameter magnitudes be required, h may in some cases be computed relatively simply analytically. 64,65 15



100 80

15

60 10 40 5 0

20 0.1 1 10 100 Relative protein concentration

0

Oligomer concentration /m*

b

20

Oligomer mass fraction / %

Average oligomer size 〈j〉

a 3

2

1

0

60 10 40 5


0


80

15



100

20

0

10 15 20 Oligomer size /j

5


5


1.5

1

0.5

0

e 80

15

60 10 40 5


0


100


f

20

0

5

d

c


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3

2

1

0

Figure 3: The key parameters controlling the globular model are the transition concentration m∗ and the transition size j ∗ . a: Average size and mass fraction as a function of protein concentration scaled by m∗ ; parameters fitted from Monte Carlo data (see main text). Systems identical apart from v0 collapse onto single curves of this form upon rescaling concentrations by m∗ . b: Size distribution at scaled p = 100 (gridline on a). Increasing Gb by 1.5× increases the average size by the same factor (c), without affecting the polydispersity significantly (d). The larger average size increases the abruptness of the oligomerization transition. Tripling Gb and h does not affect j ∗ , but increases the sharpness of the transition from small to large size (e). By reducing the concentration of large oligomers below p∗ , it also increases the value of p at which the mass fraction transition occurs relative to the size transition, but not the transition abruptness. These effects are achieved by the reduction in the polydispersity of the size distribution (f ).

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Relating oligomer and fibril formation So far we have investigated protein oligomer formation in isolation; in practice, we are frequently interested in proteins that can form both oligomers and macroscopic fibrillar aggregates, with different structures and therefore different properties. Indeed, the coupling of these processes has important implications for oligomer and fibril formation.

Oligomers as prenucleation clusters The formation of new protein fibrils from monomeric protein has been successfully modelled using a slow coarse-grained nucleation reaction step; having been nucleated, new fibrils can subsequently elongate rapidly by the addition of further monomers to the growing fibril ends. Recent research into non-classical nucleation theory has indicated that intermediate structures known as prenucleation clusters are frequently essential for nucleation to occur. 71 It has also been shown that several important types of protein fibrils nucleate via non-fibrillar oligomeric intermediate species; 50–52 these oligomers can be identified as the prenucleation clusters of non-classical nucleation theory. The globular oligomers investigated in the current work are decidedly non-fibrillar, being non-linear in geometry, and being assembled by non-directional hydrophobic interactions rather than by β-stacking. They would therefore be expected to form prior to the final stage of nucleation that produces new fibrils, being incapable of rapid filamentous growth. The non-directional nature of globular oligomer bonding makes it highly likely that they can self-assemble more rapidly than can new fibrils, whose constituent monomers require a specific structure and orientation in their constituent monomers for nucleation to occur. They are thus strong candidates for the prenucleation clusters frequently seen in protein fibril formation, holding monomers in close proximity to permit cooperativity in their restructuring into fibril-forming conformations. It is also possible that globular oligomers of the type investigated in this study could form as off-pathway intermediate species during a fibril formation reaction, with few or no new fibrils being produced from these oligomers. 17



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Relative stability of oligomers and fibrils Fibril formation by monomeric proteins involves a phase transition, below which aggregates do not form; the total monomer concentration at which this occurs is known as the critical aggregation concentration (CAC). To show why this occurs, we first define P (t) as the number concentration of fibrils of any size, and k+ and koff as the size-independent fibril elongation and depolymerization rate constants respectively. Following from earlier notation, the monomer concentration at time t is m(t). The equilibrium concentration of monomers under conditions where fibrils can form is then independent of the initial monomer concentration:

2k+ m(∞) P (∞) = koff P (∞) =⇒ m(∞) =

koff . 2k+

(14)

We may then identify m(∞) ≡ mCAC : if the initial monomer concentration is lower than this value, no macroscopic fibril formation occurs; if it is higher, mCAC becomes the equilibrium monomer concentration. Given that fibrils are expected to be thermodynamically the most stable aggregate phase, m∗ for any globular oligomers, whose negative logarithm has been shown to be the free energy per monomer in an oligomer, will typically be higher than the CAC, whose negative logarithm is the free energy per monomer in a fibril (m∗ > mCAC ). Globular oligomers will therefore never dominate over dimeric oligomers at chemical equilibrium with fibrils, no matter what the value is of the initial monomer concentration. This is borne out by studies of equilibrium Aβ oligomer concentrations; 55 which demonstrates that the predominant oligomers are dimeric.

Modelling transient oligomer formation Although we have demonstrated that globular oligomers are unlikely to be present in any significant amounts at equilibrium with fibrils, they are expected to form transiently at the much higher monomer concentrations usually present during kinetic experiments. It is often

18


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the case that the timescale of oligomerization is faster than that of fibril formation, such that a pre-equilibrium between monomers and oligomers may be a reasonable approximation. In such experiments, our equilibrium model of globular oligomer formation is therefore expected to be useful for understanding observed size distributions of oligomers, as well as for understanding fibril formation when the oligomers are on-pathway intermediates of the fibril nucleation reaction process.

Contrasting Linear and Globular Oligomers Short fibrils can be viewed as linear oligomers, and can in principle be detected by singlemolecule techniques. Additionally it is possible that linear nonfibrillar oligomers can form during the early stages of an aggregation reaction. Here we derive or present the key characteristics of linear oligomer systems based on the well-known Oosawa model. 53,54 We explain how ensembles of linear oligomers differ qualitatively from ensembles of globular oligomers (Figure 4) and how this may be taken advantage of to infer oligomer geometry from experimental studies.

Size distribution Above, we reduced the problem of finding a size distribution for an aggregating system to one of finding a suitable internal free energy form. In the case of linear oligomers, we have to a good approximation ∆G◦int (j) = (j −1)ε, where ε is the size-independent nearest-neighbour binding energy. Thus, using (1) we recover Oosawa’s result for the size distribution 53 (see Ref. 55 for further details): f (j) =

1 NA v0

e−β ε (j−1) eβ j µ .

(15)

If we instead absorb v0 into ε, identifying ε = ∆G◦+ , the size-independent free energy of monomer addition, this may be further simplified to a single-parameter model:

f (j) = e−β ε (j−1) eβ j µ . 19


(16)


We therefore expect a monotonically-decreasing equilibrium oligomer size distribution for all initial monomer concentrations, with increasing initial monomer concentration having no effect beyond increasing the slope of the size distribution in log-space (Figure 4f); this is confirmed by coarse-grained Monte-Carlo simulations of linear oligomer formation (SI Figure 1b). The size distribution for globular oligomers is qualitatively different (Figure 4b), being peaked above a critical protein concentration. Although current experimental measurements of size distributions are highly approximate, 50 this distinction is large enough that it may in certain circumstances be detected.

Critical concentration and rescaling A critical concentration analogous to m∗ , but for linear oligomers, is m∗∗ = eβε ; above this total protein concentration, most protein molecules added to the system become incorporated into oligomers. Since size distributions have dimensions of concentration, rescaling by this concentration reduces the size distribution to the dimensionless form fˆ(j) = fˆ(1)j , with fˆ(1) = f (1)/m∗∗ computable from the reduced form of the total protein concentration 53 pˆ = p/m∗∗ = fˆ(1)/(1 − fˆ(1))2 . Additionally, the reduced total oligomer concentration is given as Fˆ = fˆ(1)2 /(1 − fˆ(1)). Thus, scaling concentrations by m∗∗ causes all the key properties of linear oligomers to collapse onto single curves parameterized by pˆ; changing ε → ε + δ is equivalent to changing the concentrations on the unscaled curves p → p e−βδ (Figure 4g-h). Since m∗∗ does not correspond to a sharp transition, we expect that longer linear oligomeric species may be able to exist in equilibrium with fibrils at the end of an aggregation reaction, unlike in the globular case. This crucial difference may be taken advantage of in experiments to determine the oligomer morphologies that may be present in an aggregating system.

20


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Average size and polydispersity The average linear oligomer size may be computed with greater ease using rescaled units. We find the mean oligomer size as:

hji =

2 − fˆ(1) , 1 − fˆ(1)

(17)

which, for large pˆ, approaches pˆ1/2 . Note that this function is unbounded, unlike that for globular oligomers, which reach a well-defined maximum peak size (see Figs. 4c and 4g). We expect as a consequence that the polydispersity of the linear size distribution is much greater at high concentrations than the globular size distribution, since there is no free energy penalty for large linear oligomers. The resultant qualitative differences in average size and polydispersity may be detected comparatively easily from appropriate approximate experimental measurements of the size distribution.

21


c

b Average oligomer size 〈j〉

104 1000 100 10

Increasing protein concentration

1 0.1 2

4

6 8 10 12 14 Oligomer size / j

12 10 8 6 4 2 0

0.100 0.010

10- 4

1 10 100 1000 Protein concentration / μM

g

f

0.001

Decreasing v0

Increasing protein concentration

2

4 6 8 Oligomer size / j

10

12 10 8 6

Decreasing ΔG○

4 2 0



105


.

e

Oligomer concentration / μM

Oligomer concentration / nM

a

...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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100

d

80 60

Decreasing v0

40 20 0

100


h

80 60 40 20 0

Decreasing ΔG○


Figure 4: The differing properties of globular (a-d) and linear (e-h) oligomers. Whereas globular oligomers are expected to form from disordered polypeptide chains in amphiphilic conformations (a), linear oligomers may be formed from more structured proteins with two discrete bonding regions (e). Whereas globular oligomer size distributions become peaked at sufficiently high total protein concentration p (b), linear size distributions always decrease exponentially (f ), simply falling off less steeply with increasing j at higher values of p. Average globular oligomer size (c) transitions steeply from hji = 2 to a well-defined higher value when p is increased sufficiently beyond p∗ ; the linear oligomer average size increases more gradually with rising p, but without bound (g). As a consequence, the linear oligomer mass fraction (h) also increases more gradually than that of the globular oligomers (d). Decreasing v0 has the effect of rescaling globular concentrations to higher values, without changing the phase-like properties (c-d; dashed lines); the equivalent parameter in the linear case is the sole model parameter β∆G◦ (g-h; dashed lines). Scaling linear system monomer ◦ and oligomer concentrations by m∗∗ = eβ∆G therefore simplifies the model by nondimensionalizing it just as scaling globular system concentrations by m∗ does.

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Conclusions In summary, we have determined that a theory of micelle formation is suitable for modelling the formation of globular oligomers with comparatively low aggregation numbers from simple amphiphilic protein monomers, demonstrating that this approach can reproduce coarsegrained Monte-Carlo simulations of oligomer formation from such monomers. As a result, similarly to larger traditional micelles, when the total protein concentration p is below a critical value p∗ , the oligomer size distribution monotonically decreases. Increasing p above p∗ , however, the distribution is soon dominated by a peak at higher aggregation number, and soon after most protein molecules rapidly become incorporated into oligomers. We have found that a particularly convenient non-dimensional form of the model could be obtained by scaling all concentrations by the characteristic monomer concentration m∗ associated with p∗ . This could be expressed in terms of the fundamental observables of the model, revealing the crucial scaling laws and simplifying the analysis of experimental data. We also developed bounds for the average globular oligomer size for when p > p∗ , finding that the globular size tends asymptotically towards a limiting value. We have also shown that the presence of fibrils suppresses globular oligomer formation at equilibrium, such that globular oligomers will not be present at the completion of a fibril aggregation reaction. Finally we contrasted the properties of globular oligomers with those of linear oligomers, highlighting the key distinctions that may be looked for in experiments designed to reveal oligomer morphology. The most notable features are the lack of a peak in the linear size distribution, which is instead monotonic decreasing with aggregation number j, and the possibility for larger linear oligomers to coexist with fibrils at equilibrium. Although the monomers in our model are simpler than the disordered polypeptide chains comprising globular protein oligomers observed in experimental studies, we believe this modelling approach has value for understanding the qualitative features of such oligomers. Moreover, since some of the parameters of the model may in principle be computed at least approximately ab initio, it may prove possible to rationalize some quantitative properties of 23



protein oligomers using this theory. These insights should contribute to the development of a detailed understanding of the mechanism of the early steps in the aggregation reactions that ultimately generate amyloid fibrils in misfolding diseases.

Acknowledgements We acknowledge support from the Schiff Foundation (AJD), the Royal Society (AŠ), the Academy of Medical Sciences and Wellcome Trust (AŠ), Peterhouse, Cambridge (TCTM), the Swiss National Science foundation (TCTM), the Wellcome Trust (TPJK), the Cambridge Centre for Misfolding Diseases (TPJK), the BBSRC (TPJK), the Frances and Augustus Newman foundation (TPJK). The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) through the ERC grant PhysProt (agreement n◦ 337969). We thank Daan Frenkel for several useful discussions.

Supporting Information The Supporting Information (SI) file contains a more detailed explanation of the statistical mechanical formulation used for oligomer size distributions, a verification that steric effects between hydrophilic segments at the micelle surface may be neglected in front of the j 5/3 connectivity-enforcing energy penalty, a figure showing the match between the linear Monte Carlo simulations and the linear statistical mechanical model, and finally derivations for the critical concentration m∗ and critical size j ∗ .

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Hydrophilic segment Hydrophobic segment

Misfolded proteins

Flexible amyloid oligomers

Fraction of oligomers / %

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Monte Carlo simulations

12 10 8 6 4

Statistical mechanical model

2 0

2


TOC Graphic

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Statistical Mechanics of Globular Oligomer Formation by Protein

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