Explaining the length threshold of polyglutamine aggregation

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2012 J. Phys.: Condens. Matter 24 244105

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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 244105 (5pp) doi:10.1088/0953-8984/24/24/244105

Explaining the length threshold ofpolyglutamine aggregation

Paolo De Los Rios1, Marc Hafner2 and Annalisa Pastore3

1 Laboratory of Statistical Biophysics, ITP SB EPFL, CH-1015 Lausanne, Switzerland2 School of Computer and Communication Sciences, Ecole Polytechnique Federale de Lausanne(EPFL), CH-1015 Lausanne, Switzerland3 Division of Molecular Structure, Medical Research Council, National Institute for Medical Research,NW7 1AA London, UK

E-mail: paolo.delosrios@epfl.ch

Received 31 October 2011, in final form 3 April 2012Published 18 May 2012Online at stacks.iop.org/JPhysCM/24/244105

AbstractThe existence of a length threshold, of about 35 residues, above which polyglutamine repeatscan give rise to aggregation and to pathologies, is one of the hallmarks of polyglutamineneurodegenerative diseases such as Huntington’s disease. The reason why such a minimallength exists at all has remained one of the main open issues in research on the molecularorigins of such classes of diseases. Following the seminal proposals of Perutz, most researchhas focused on the hunt for a special structure, attainable only above the minimal length, ableto trigger aggregation. Such a structure has remained elusive and there is growing evidencethat it might not exist at all. Here we review some basic polymer and statistical physics factsand show that the existence of a threshold is compatible with the modulation that the repeatlength imposes on the association and dissociation rates of polyglutamine polypeptides to andfrom oligomers. In particular, their dramatically different functional dependence on the lengthrationalizes the very presence of a threshold and hints at the cellular processes that might be atplay, in vivo, to prevent aggregation and the consequent onset of the disease.

(Some figures may appear in colour only in the online journal)

1. Introduction

The neurodegenerative Huntington’s disease is caused bycell death due to the aggregation of polyglutamine repeats(polyQs) contained in the sequence of the exon-1 of theprotein hungtintin. The age of onset of the disease correlateswith the length of the polyQ tracts: the longer the repeat, thesooner the symptoms appear, although aggregation and theassociated pathologies are observed only above a thresholdof 35–40 residues [1, 2]. The structural hypothesis, eminentlyput forward by Perutz, addresses the presence of the lengththreshold by positing that a minimal number of glutamineresidues are necessary for the polyQ peptide to adopt aspecific conformation that would then be aggregation-prone,possibly via a nucleation mechanism, by self-assembling intoordered structures [3]. Although such structures, and the

oligomers, do indeed become more stable as the numberof residues increases [4], it has also been shown that theiroverall stability depends on the timescale of the simulations,with longer trajectories hinting at thermodynamically unstableconstructs [5]. Moreover there are strong indications thatsingle polyQ monomers in solution are compact andunstructured [6], or possibly random coils [7]. Unfortunately,the quest for the elusive structure that might seed aggregationhas, till now, overshadowed some simple considerations thatcan rationalize the presence of a length threshold. Here weuse basic polymer and statistical physics principles to showhow the length dependence of the association and dissociationconstants can modulate the aggregation propensities of polyQso that the solution is mostly populated by monomers orsmall oligomers for lengths below the threshold, and by largeaggregates above it. Most importantly, our conclusions do

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J. Phys.: Condens. Matter 24 (2012) 244105 P De Los Rios et al

not depend on the shape of aggregates and on the particularabundance of secondary structures, most likely β-sheets,that they exhibit. Such features determine the values of theparameters of our model, but not its overall structure.

2. Some simple preliminary considerations

Because in general the free energy of any system governedby short range interactions is extensive with the number ofinteracting degrees of freedom, the binding free energy ofa polyQ monomer composed of n monomers (henceforthreferred to as Qn) to another Qn monomer, or to anoligomer made of m monomers (m-mer), is proportional to n:1G = −εn (−ε being the binding free energy per residue).Consequently the dissociation rate decreases exponentiallywith n (n ≥ 1), as from the Kramers formula [8]:

kesc(n) = kesc,0e−εn

kBT (1)

(kB is the Boltzmann’s constant and T the absolutetemperature; kesc,0 has the dimensions of a rate). The bindingof Qn monomers to each other to form dimers, or to m-mers toincrease their size (m→ m+1), is instead a two-step process:first, the two molecules must encounter by diffusion, thenthey must bind. Typically, the second step is faster than thefirst one and the binding reaction is limited by the encounterrate: kenc(m)c1cm, where c1 and cm are the concentrations ofmonomers and m-mers, respectively; kenc(m) is the encounterconstant, which depends on the oligomer size m but not onn. The dissociation constant of monomers from m-mers isthus, to a first approximation, Kd(n,m) = kesc(n)/kenc(m) =Kd0(m) exp(−εn/kBT). In this formula it is reasonable toexpect some dependence of the binding energy ε on the sizeof the oligomer, since larger oligomers provide more chancesfor individual residues to coordinate with others, althougholigomers whose linear size is larger than the one of a singlepolyQ monomer are likely to be characterized by the samevalue of ε, irrespective of their size, because they all look likeinfinite surfaces for the incoming monomer.

Our main observation is that, when the binding ratedominates over the unbinding one, kenc > kesc, aggregates cangrow. The relation between the kinetic constants translatesinto a thermodynamic relation between the total Qn monomerconcentration and the dissociation constant, namely c1 >

Kd(n) (here for simplicity we disregard the dependence on mof the encounter rate). Using a polyQ with 20 residues (Q20)as a reference point, Kd(n) = Kd(20) exp[−(ε/kBT)(n−20)],the above inequality is satisfied if

n > nthreshold = 20+ (kBT/ε) ln[Kd(20)/c1]. (2)

The observation that Q20s do not aggregate at physiologicalconditions at 1 µmol concentrations suggests that Kd(20) >1µmol [9]. Since glutamine is mildly hydrophilic, the bindingfree energy per monomer ε is at most of the same order asthe thermal energy (kBT/ε ' 1) [10], so that we can inferthat a lower bound for the aggregation threshold is 20 −ln(c1/1 µmol).

In the next sections we make the above arguments morerigorous by following a three-pronged approach. First we

Figure 1. Free energy profile (blue line) and internal energy profile(red line) for two Q24 as a function of the distance x between theircenters of mass. The barrier height is 1G and the distance x0determines when the two polymers can be considered as not beingin contact anymore.

derive, by way of Monte Carlo simulations, the dependenceof the free energy profile of unbinding on the length ofthe polymer. Next, using such a landscape we computethe relevant kinetic encounter and escape rates. Finally, weinvestigate the effects of the length dependence of the rateson aggregation by using a scheme based on rate equationsdescribing the formation of progressively larger aggregates.

3. Estimate of the escape rate

To calculate the dissociation constant of an aggregate madeof two polyQs, we performed Monte Carlo simulations ofsystems with two polymers. The polyQ chains are modeledas freely jointed chains with self-avoiding beads. The length nof the chains corresponds to the number of beads. The radiusof the beads is r0 = 2.5 A and they are 3.8 A apart [11].A Lennard-Jones potential describes the interaction betweenpairs of non-consecutive beads, with a minimal value of V0 =

−kBT300 K ≈ 4.14 × 10−21 J, corresponding to the attractionenergy of two glutamines [10].

To study the stability of a system with two chains,we calculated the free energy with respect to the distancex between the centers of mass of the two polymers usedas a reaction coordinate. To calculate the free energyprofile we performed Monte Carlo simulations with theMetropolis algorithm [12]. To improve the sampling in thehigh energy regions of the landscape, we used restrainingpotentials along the reaction coordinate and used the weightedhistogram analysis method (WHAM) [13] to extract theoverall landscape. The result is an energy profile F(x) that hasa minimum for small x which corresponds to the aggregatedstate (figure 1, n = 24). The height of the energy barrier(1G) for two chains grows linearly with n, as expected bythe extensivity of the free energy (figure 2, upper panel).

By defining x0 as the minimal distance where the twochains do not touch each other (internal energy close to 0,see the red line in figure 1), it is then possible to compute the

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J. Phys.: Condens. Matter 24 (2012) 244105 P De Los Rios et al

Figure 2. The linear dependence of 1G (as determined in figure 1)on the polyglutamine length n is clearly discernible in the upperpanel, and it confirms the expectations based on the simpleextensivity of the free energy. The exponential dependence of theescape rate on n (lower panel) as calculated from equation (3)confirms the validity of the simple approximation (1).

expected escape rate using the Kramers formula [8]

kesc(n) = k0

[∫ x0

0e−βFn(x)

(∫ x0

xeβFn(y) dy

)dx

]−1

(3)

with β = 1kBT (T = 300 K) and k0 a constant that depends

on temperature and on the effective friction along thereaction coordinate. Figure 2 (lower panel) shows that kesc(n)decreases exponentially with n, so that the above-mentionedformula (1) is a reasonable approximation. We have thusconfirmed our original expectations stemming from basicconsiderations about the intuitive extensivity of the freeenergy.

4. Estimate of the encounter rate

The rate of dissociation is counterbalanced by the associationrate kenc that can be estimated to a first approximation usingSmoluchowski theory [14]. According to Smoluchowski, twoparticles of radii R1 and R2, moving at random with diffusionconstant D1 and D2, respectively, encounter at a rate

kenc = 4π(D1 + D2)(R1 + R2)c1c2 (4)

where D1 (D2) is, by Einstein’s relation, D1 = kBT/(6πηR1)

(D2 = kBT/(6πηR2)), η being the solvent (water) viscosity(0.1 cp) and c1 and c2 the concentrations of the two kinds ofparticles.

In the present context the two particles are a Qn monomerand an m-mer. We describe them as spherical objects ofradii Rm = [3/(4π)mn]1/3r0 and RQn = [3/(4π)n]

1/3r0, thelatter consistent with the observation that polyQ monomersare in compact but unstructured conformations [6]. As aconsequence the encounter rate is

kenc(m, n) = 4π(Dm + DQn)(Rm + RQn)

=23

kBT

η

(1

m1/3 + 1)(m1/3

+ 1)c1cm (5)

which does not depend on n anymore as it cancels out of theequation.

5. Aggregation reaction equations

The functional forms of the encounter and escape ratesobtained above allow us to devise a set of rate equationsdescribing the aggregation process as the addition (subtrac-tion) of monomers from m-mers to form (m+ 1)-mers ((m−1)-mers):

dc1

dt= −2kenc(1, n)c2

1 −

mmax−1∑j=2

kenc(j, n)c1cj

+ 2kesc(n)c2 +

mmax∑j=3

kesc(n)cj · · ·

dcm

dt= −kenc(m, n)c1cm + kenc(m− 1, n)c1cm−1

− kesc(n)cm + kesc(n)cm+1 · · ·

dcmmax

dt= kenc(mmax − 1, n)c1cmmax−1 − kesc(n)cmmax

(6)

where ck(t) is the concentration of k-mers at time t.By dividing all equations by kesc(n) (which is tantamount

to a rescaling of time) the equations become

dc1

dτ= −2K−1

d (1, n)c21 −

mmax−1∑j=2

K−1d (j, n)c1cj

+ 2c2 +

mmax∑j=3

cj · · ·

dcm

dτ= −K−1

d (m, n)c1cm + K−1d (m− 1, n)

× c1cm−1 − cm + cm+1 · · ·

dcmmax

dτ= K−1

d (mmax − 1, n)c1cmmax−1 − cmmax

(7)

The dissociation constants are then written with respectto Kd(1, 20):

Kd(m, n) = 4Kd(1, 20)[(

1

m1/3 + 1)(m1/3

+ 1)]−1

× e−ε

kBT (n−20). (8)

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J. Phys.: Condens. Matter 24 (2012) 244105 P De Los Rios et al

Figure 3. Thermal plots of the fractions of monomers in oligomers of different sizes, as a function of the oligomer size and of the monomerlength. As usual, larger values are colored red, smaller ones are colored blue. (A) Dependence of the fractions upon the largest oligomer size(30, upper panel, and 50, lower panel). Clearly, below the threshold the fractions perfectly overlap, the only difference being above thethreshold where the monomers concentrate within the largest size oligomers. (B) The position of the threshold only mildly changes uponlarge variations of the dissociation constant (0.5 µmol, upper panel, and 50 µmol, lower panel), shifting by only about five residues, whichis close to the logarithm of the dissociation constant ratio, as predicted in equation (2).

Ideally mmax → ∞ but for computational reasons we set itto a fixed value and then perform the numerical integrationfor different values of mmax to ensure that the results do notdepend on its precise value. Equations (7) correctly conservethe quantity c1,tot =

∑jjcj which is the total amount of

monomers in solution.In figure 3(A) we show the results for the integration

of (7) with c1,tot = 1 µmol and Kd(1, 20) = 5 µmol formmax = 30 (upper panel) and mmax = 50 (lower panel).We plot the thermogram corresponding to the fraction ofmonomers contained in oligomers of size m: mcm/

∑k(kck) =

mcm/(c1,tot). Visibly, below the threshold (black lines)the equilibrium concentrations are not affected by thevalue of mmax, whereas above the transition the monomersare localized in the largest possible oligomers, of sizeproportional to mmax, implying that above the threshold thelargest oligomer size would diverge if mmax →∞.

In figure 3(B) we show the result for the integration of (7)with c1,tot = 1 µmol and Kd(1, 20) = 0.5 µmol (upper panel)and Kd(1, 20) = 50 µmol for mmax = 30 (lower panel). Theposition of the threshold (black line) only changes by aboutfive residues, which is, as predicted, close to ln(50/0.5) = 4.6(see equation (2)).

6. Conclusions

The presence of a threshold length for polyglutamineaggregation is one of the defining features of Huntington’sdisease (and of several other polyQ expansion pathologies),and its origin has been one of the main open issues in researchon neurodegenerative diseases at the molecular level. Theoriginal proposal by Perutz and co-workers that a minimallength should be necessary to form special aggregation-pronestructures has not found, in more recent times, experimentalor theoretical support. Rather, there is growing evidencethat the length of the polyglutamine repeats modulates thekinetics of the aggregation process [15]. Here we haveshown that the dramatic difference in the functional formsof the encounter and escape rates of polyQ monomers fromoligomers, the former weakly, if at all, dependent on n,the latter decreasing exponentially, naturally gives rise to alength threshold. The precise value of the threshold lengthdepends instead on the solution conditions, the temperatureand the basal concentration of polyQ monomers (as fromequation (2)) [16]. In this respect it is worth guessingwhat a reasonable physiological polyQ concentration couldbe. Given an average neuron radius of 20 µm, the neuronvolume is about 30 000 µm3 (spherical approximation) and

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J. Phys.: Condens. Matter 24 (2012) 244105 P De Los Rios et al

a 1 µM concentration would turn into about 107 polyQmolecules. Should polyQ-containing proteins be low ormoderate copy-number proteins, more realistic concentrationswould be in the nM range and, as a consequence, ourtreatment would predict a threshold at about 29 residues,closer to the physiological values. Yet rates and thresholdin vivo are modulated by other factors, such as the presenceof chaperones that might further reduce the concentrationof aggregation-prone polyQ monomers by holding them,or that might directly scavenge aggregates (thus increasingthe effective dissociation rate) [17–19]. Also the proteincontext of the polyQ repeats and the cellular crowdinglevel are surely going to affect the precise values of therates entering equations (7) [20]. Given our insufficientquantitative knowledge of each of these cellular processes,and their likely variability during aging, it might be difficultto correlate the precise value of the threshold length betweenin vivo, in vitro and in silico studies, whereas the simplekinetic principles outlined here can rationalize the verypresence of the threshold. Due to the paradigmatic roleof polyQ for poly-aminoacid and misfolding diseases, ourobservations are likely to be relevant well beyond the specificpolyglutamine case. For example, polyalanine repeats arealso known to aggregate only above a length threshold ofabout 20 residues [21], shorter than the one typical of polyQbecause of the hydrophobic nature of alanine. Furthermore,our results show that the dependence of the kinetic rates on thelength of the aggregating peptides strongly affects the systembehavior and therefore should be duly taken into account incomputational approaches to aggregation by means of rateequations, a strategy that is finding growing interest [22, 23].

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