The role of clearance mechanisms in the kinetics of toxic protein aggregates involved in neurodegenerative diseases
Travis B. Thompson, Georg Meisl, Tuomas Knowles, Alain Goriely
TThe role of clearance mechanisms in thekinetics of toxic protein aggregates involved inneurodegenerative diseases.
Travis B. Thompson ∗ , Georg Meisl † , Tuomas Knowles † , ‡ , and Alain Goriely ∗∗ Mathematical Institute, Andrew Wiles BuildingWoodtsock Rd University of Oxford OX2 6GG, UK † Centre for Misfolding Diseases, Department of Chemistry,University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK ‡ Cavendish Laboratory, University of Cambridge,19 JJ Thomson Avenue, Cambridge CB3 0HE, UK
September 30, 2020
Abstract
Protein aggregates in the brain play a central role in cognitive decline and structural damageassociated with neurodegenerative diseases. For instance, in Alzheimer’s disease the formationof Amyloid-beta plaques and tau proteins neurofibrillary tangles follows from the accumulationof different proteins into large aggregates through specific mechanisms such as nucleation andelongation. These mechanisms have been studied in vitro where total protein mass is conserved.However, in vivo, clearance mechanisms may play an important role in limiting the formation ofaggregates. Here, we generalize classical models of protein aggregation to take into account bothproduction of monomers and the clearance of protein aggregates. Depending on the clearancemodel, we show that there may be a critical clearance value above which aggregation does nottake place. Our result offers further evidence in support of the hypotheses that clearance mech-anisms play a potentially crucial role in neurodegenerative disease initiation and progression;and as such, are a possible therapeutic target.
Alzheimer’s disease (AD), and other related neurodegenerative diseases, are associated with theassembly of specific, toxic proteins into fibrillar aggregates. Alzheimer’s disease, in particular, ischaracterized by the aggregation of Amyloid- β (A β ) plaques and tau protein neurofibrillary tangles(NFT). The role of A β in Alzheimer’s is thought to be so central to the disease that it is the basisof the so-called ‘Amyloid- β hypothesis’ [1, 2, 3], stating that that the accumulation and depositionof oligomeric or fibrillar amyloid beta peptide is the main cause of the disease. This hypothesishas provided a guide for most of AD research over the last 20 years. However, recent experimental1 a r X i v : . [ q - b i o . B M ] S e p vidence, and the failure of several drug trials, has lead to renewed scrutiny of this foundationalassumption.The production of A β is a natural process related to neuronal activity. Indeed, A β is a nor-mal metabolic waste byproduct [4, 5] that is typically removed from intracellular and extracellularcompartments by several clearance mechanisms [6, 7]. In healthy subjects waste proteins are bro-ken down by enzymes, removed by cellular uptake, or efflux to cerebrospinal fluid compartmentswhere they eventually reach arachnoid granulations, or lymphatic vessels. While healthy clearancemechanisms, working in harmony, avert the buildup of toxic A β plaques and tau NFT; their im-pairment or dysfunction can lead to toxic levels of aggregates [7]. The specifics of in-vivo clearancemechanisms remain a topic of clinical debate; however, the kinetics enabling proteins to amass intotoxic aggregates can be carefully, and systematically, studied in vitro and under varied conditions.The production of A β , at a high level, is mediated by a membrane protein called amyloid precursorprotein (APP). APP is typically cleaved by α -secretase and the resulting products do not aggre-gate. However, APP can also be cleaved by β -secretase, which results in soluble monomeric APPfragments of different sizes. The most common size categories are A β
38, A β
40, and A β
42. Whilemonomeric A β
38 is not prone to further aggregation; A β
40 and A β
42, containing two additionalamino acids at the C terminus, are the main isoforms of interest in the study of AD pathology.Protein aggregation pathways are, in general, complex and involve multiple steps. In fact, it hasrecently been shown [8] that the aggregation properties of A β
40, which is more abundant, differ fromthose of the more aggregate-prone A β
42; even under the same conditions. A theoretical frameworkof chemical kinetics and aggregation theory [9, 10, 11] has been combined with careful, systematicin vitro experiments performed under differing conditions; such as varied concentration or pH.This approach has: elucidated effective pathways and mechanisms for nucleation, aggregation andfragmentation [12]; and produced a deep understanding of key properties, underlying the formationof aggregates under ideal conditions, with the potential for therapeutic intervention [13, 14].Here, we develop a mathematical framework to describe the effects of clearance and monomerproduction chemical kinetics driving aggregation; we apply the framework to the study of A β . Toaccomplish this, we extend the current theory describing A β aggregation in vitro, which has beenvalidated against experiment, to include monomer production and oligomer clearance terms. Inparticular, we study two different clearance mechanisms: one where total mass is conserved (size-independent clearance); and one where it is not (size-dependent clearance). In the former case weshow the full system reduces to three equations amenable to a systematic analysis. We identify acritical value of clearance above which the production of toxic aggregates does not take place. Ourresults offer further evidence in support of two main hypotheses: that clearance mechanisms playa crucial role in neurodegenerative disease initiation and progression; and that therapies enhancingclearance above a prescribed, critical value may serve as a possible intervention strategy. In partic-ular, we will exhibit the existence of critical clearance values; such values are consistent with theobservation of disease onset when natural clearance mechanisms within the brain have degradedthrough aging. Our model for protein aggregation-dynamics model includes multiple mechanisms: heterogeneousprimary nucleation; homogeneous primary nucleation; secondary nucleation; linear elongation; frag-2entation; and clearance (c.f. Fig. 1). These mechanisms lead to a general class of mathematicalmodels that can describe a wide range of aggregating systems in vitro. In particular, by includ-ing heterogeneous primary nucleation terms, a source term for new nuclei, that is independent ofmonomer concentration, is present; this source is in addition to the usual monomer-dependent ho-mogeneous primary nucleation. Thus, in such a model, the importance of interfaces in the initiationof nucleation is sufficiently accounted for. In the model, each aggregate of a given size is representedby a population. In general, each population, with aggregates of size i , will be represented by anindexed concentration; we use the special notation m ( t ) for the monomer population i = 1 , whileall other aggregate concentrations are denoted by p i ( t ) for i ∈ { , , . . . } . The master equations arethen: d m d t = γ − λ m − k − n c k n m n c − k + mP − n k σ ( m ) M + 2 k off P (1)d p d t = − λ p + k + δ ,n c k n m n c − k + mp + δ ,n k σ ( m ) M + 2 k off p (2)d p i d t = − λ i p i + δ i,n c k n m n c + 2 k + m ( p i − − p i ) + 2 k off ( p i +1 − p i ) + δ i,n k σ ( m ) M, i > , (3)where δ i,j is Kronecker’s delta (1 if i = j and 0 otherwise) and σ ( m ) = m n K M K M + m n , P = ∞ (cid:88) i =2 p i , M = ∞ (cid:88) i =2 ip i . (4)Here, P and M are the first two moments of the population distribution; they represent the totalnumber and total mass of aggregates, respectively. In these equations, the parameters representthe following effects, sketched in Fig. 1: γ : (constant) monomer production such as by β -secretasemitigated cleavage of APP, driving mass influx; λ i : clearance of aggregate of size i such as bylymphatic or cellular processes; k : heterogeneous primary nucleation (independent of the monomerconcentration); k n and n c : nucleation of aggregates of size n c > ; k + : linear elongation transformingaggregate from size i to i + 1 ; k : secondary nucleation of aggregates of size n > ; K M : saturationof the secondary nucleation; k off : depolymerization by one monomer.The aggregation model (1)-(3), and its many variations, have served as a template for in vitroexperiments [8, 13, 15]. Multiple experimental fittings have shown that the exponents n c and n are: n c = n = 2 for A β
40 and for A β
42 in the presence of a PBS buffer [8, 15]; for A β
42 inthe presence of a HEPES buffer n = 2 and n c = 0 provides the best fit [13]. For the discussionsand derivations in this manuscript we take the view of PBS buffer experiments [8, 15] so that n c = n = 2 . Adaptation to A β
42 HEPES, so that n c = 0 , is straightforward and all numericalresults are qualitatively similar. When fitting experimental data it is often the case that only one of k or k n , depending on the best data fit, is used; i.e. that either heterogeneous primary nucleationor homogeneous primary nucleation best explains the particular experimental data.The primary purpose of this manuscript is to describe the qualitative impact of clearance mech-anisms in the dynamics and a particular choice of nucleation mechanism, i.e. k = 0 versus k n = 0 ,does not affect the results. Examples of fitted A β model parameters are listed in Table 1. PBSand HEPES refer to the buffers used in the corresponding experiments. Aggregation, in the fittedexperiments, proceeds much faster than depolymerization and k off = 0 is found to be a good fit todescribe the dynamics. However, from a theoretical point of view, we note that k off = 0 implies that3igure 1: Mechanisms included in the master equations (1)-(3). We consider multiple effects forthe formation of aggregates into our systems with rates constants k i . The constants correspondingto transfer of mass to and from the external system are represented by greek letters ( γ and λ i ).The process of heterogeneous nucleation (with constant k ) is similar to homogeneous primarynucleation and is not depicted (the main difference being that its rate does not depend on themonomer concentration).there is no non-vanishing stationary distribution in the absence of clearance and production terms.Here, we will first follow experimental data and take k off = 0 . Then, we will show that the addi-tion of this small term does not change our results. Therefore, we will use the fitted experimentalparameters given in Table 1. Clearance and production have not been investigated experimentally;thus, we leave them as free parameters. In particular, we will be interested in determining particularvalues of these parameters when a qualitative change of the dynamics occurs. In the case of size-independent clearance, we have λ i = λ > for all i . Our main question isto understand the role of the clearance term. In particular, we will establish that if clearance issufficiently large, the formation of aggregates does not take place.4able 1: Typical parameters for the Aβ model. PBS and HEPES refer to the buffer used for theexperiments. Note that A β
42 is generally faster than A β
40 and HEPES buffer is faster than PBS.In these experiments, aggregation is sufficiently fast so that k off = 0 provides a good fit. The valuesof λ crit give the critical values of clearance and their approximations for the size-independent case(see text). The values of τ and τ give the typical time scales associated with each dynamics (seetext).param. mechanism A β
40 PBS [15] A β
42 PBS [8] A β
42 HEPES [16] units k heterogeneous nucleation 0 0 . × − M h − k n homogeneous nucleation . × − . × − − n c h − n c homogeneous nucleation 2 2 2 unitless k secondary nucleation . × . × . × M − h − n secondary nucleation 2 2 2 unitless K M saturation . × − . × − . × − M k + elongation . × . × × M − h − k off depolymerization 0 0 0 h − m Initial monomer c. × − × − × − M λ crit critical clearance 0.72 2.45 17.0 h − ˜ λ crit perfect bifurcation 0.72 2.47 17.0 h − α nonlinear coefficient 312,042 647,390 2.83726 × M − h − τ exponential time scale 1.4 0.4 0.06 h τ amplification time scale 12.6 2.5 0.4 h λ (1) crit critical clearance ν = 1 × − × − × − h − λ (0) crit critical clearance ν = 0 − λ ( − crit critical clearance ν = − × × × h − In the size-independent case, a well-known but remarkable feature of the system (1)-(3) is that aclosed system of equations for the first two moments P and M and the monomer concentration m = p can be obtained exactly:d P d t = − λP + k + k n m + k σ ( m ) M, (5)d M d t = − λM + 2 k + 2( k + m − k off ) P + 2 k n m + 2 k σ ( m ) M, (6)d m d t = γ − λm − k − k + m − k off ) P − k n m − k σ ( m ) M, (7)where σ ( m ) = m K M / ( K M + m ) and we have chosen n = 2 . The total mass of the system M tot = M + m satisfies, by summing (6)-(7), the evolution equationd M tot d t = − λM tot + γ. (8)This equation implies that the total mass in the system evolves to a stable steady state M tot = γ/λ with a typical time-scale /λ . To simplify the analysis, we will further assume that, initially, the5ystem is at this state by choosing the following unseeded initial conditions M (0) = P (0) = 0 , m (0) = m = γ/λ, (9)and the total mass of the system is conserved for all time M tot ( t ) = m . The term ‘unseeded’ refersto the fact that, initially, there is no toxic protein in the system (hence, no seed). This conditionassumes a lack of aggregated species in a healthy in vivo state. Indeed, it is observed that solubleA β monomers are found in healthy individuals of all ages while aggregates larger than monomersare correlated with Alzheimer’s disease progression [17]. An extra advantage of this approach isthat it fixes the constant γ = m λ .Before we study the system in full generality, it is useful to consider the overall dynamic ofthe system for a typical set of parameters for the aggregation of A β
40 given in the first column ofTable 1. We will use this set of parameters for all our examples. The other data sets are qualitativelyequivalent and the values of various derived quantities are given in Table 1. As shown in Fig. 2,the typical behavior of the system from an unseeded initial condition is for the toxic protein massto increase up to finite value M ∞ while the monomer concentration decreases to m ∞ in a typicalsigmoid-like behavior. We observe that, in the absence of clearance, the monomer population isFigure 2: Typical dynamics of the monomer (blue) and toxic (red) concentration (in moles) fordifferent values of the clearance ( λ in h − ) for A β
40 with parameters from Table 1 and λ = 0 (largedashed), λ = 0 . (solid) and λ = 1 (small dashed). Asymptotic values for λ = 0 . are shown withdotted lines.completely converted to toxic proteins ( λ = 0 , dashed curves in Fig. 2). Conversely, for largeclearance almost no conversion takes place ( λ = 1 , dotted curves in Fig. 2). Some of the monomersare converted (solid curves for λ = 0 . in Fig. 2) for the case of moderate clearance. Of particularinterest for our discussion is the change of behavior at some critical value λ crit of the clearance λ where aggregation becomes negligible. 6o derive an exact value for λ crit , we determine the dependence of the asymptotic states m ∞ on λ . Using the steady state hypothesis with m = m ∞ , P = P ∞ and M = M ∞ in (5)-(6) one expressesthe latter two states as a function of the parameters, λ and m ∞ . These relations are substituted in(7) to produce the implicit equation q ( λ, m ∞ ) = 0 with q ( m ∞ , λ ) = 2 k + k n m ∞ − m ∞ (2 k + k K M − λk n + 2 k n k off ) − m ∞ (cid:0) − λ − k + k m K M + 2 k λK M − k + k n K M − k k off K M − k + k (cid:1) − m ∞ (cid:0) − k λ − k λm K M + 2 k m k off K M − λk n K M + 2 k n k off K M + 2 k k off + λ m (cid:1) + m ∞ (cid:0) λ K M + 2 k + k K M (cid:1) + 2 λk K M − k k off K M − λ m K M (10)For instance, for the same parameter values as in Fig. 2, we show in Fig. 3 the values of m ∞ asa function of λ . We observe a sharp transition for a critical value of the clearance parameter λ .Figure 3: Perfect (red) and imperfect (blue) transcritical bifurcation obtained for A β
40. Unstable(dashed) and stable(solid) equilibrium solutions. In this case, we have ˜ λ crit ≈ . and ˜ m ∞ ≈ . . λ . Dashed curves indicate unstable equilibria solutions and solid curves denote stableequilibria.There are three necessary conditions for λ crit : first that λ crit is non-negative; second that m ∞ ismaximal; and third that the value of m ∞ coincides with m . The last two conditions can be realizedby computing the derivative of the expression q ( λ, m ∞ ) = 0 evaluated at m = m ∞ . Therefore, λ crit is given by the positive root of L ( λ ) = 0 where L ( λ ) = ∂q∂m ∞ (cid:12)(cid:12)(cid:12)(cid:12) m ∞ = m = − k off K M m ∞ ( k m + k n ) + 6 k + k n K M m ∞ + 6 k k off K M m ∞ − k + k K M m ∞ + 6 k + k m K M m ∞ + 2 k + k K M − k n k off m ∞ + 10 k + k n m ∞ − k k off m ∞ + 6 k + k m ∞ + λ (cid:0) K M m ∞ ( k m + k n ) − k K M m ∞ + 8 k n m ∞ + 4 k m ∞ (cid:1) + λ (cid:0) K M + 3 m ∞ − m m ∞ (cid:1) (11)For A β -40 the critical clearance, as shown in Fig. 3, is λ crit = 0 . . Critical clearance rates for theother experimental data sets are given in Table 1 for comparison.7 .2 Bifurcation and normal form analysis In a neighborhood of λ crit , m ∞ , as a function of λ , undergoes a sharp transition. This transition isnot a bifurcation in the strict sense but, in the parlance of dynamical systems, it can be described asan imperfect transcritical bifurcation when heterogeneous nucleation and homogeneous nucleationterms can be understood as an imperfection and are sufficiently small with respect to the elongation.More specifically, when k / ( k + m ) (cid:28) and k n /k + (cid:28) the system is well approximated by k = 0 and k n = 0 . In this limiting case, the fixed point ( P, M, m ) = (0 , , m ) for the system (5)-(7) undergoes a (perfect) transcritical bifurcation at ˜ λ crit ≈ λ crit that can be obtained by locallyexpanding m ∞ in λ to find ˜ m ∞ = m + 1 α ( λ − ˜ λ crit ) + O (cid:16) ( λ − ˜ λ crit ) (cid:17) , (12)where ˜ λ crit is specified by the formula ˜ λ crit = m (cid:16)(cid:113) k K M (cid:0) m ( k m + 2 k + ) K M − k off K M + 2 m ( k + m − k off ) (cid:1) + k m K M (cid:17) K M + m , (13)and α is defined by the expression α = m (cid:16) k K M (cid:16) λ crit + 3 k + m − k off (cid:17) − ˜ λ crit (cid:17) ˜ λ crit (cid:0) K M + m (cid:1) − k m K M . (14)When the clearance is close to the critical value the linear approximation to the perfect bifurcationis a reasonable approximation for the imperfect bifurcation as can be appreciated in Fig. 3 where ˜ λ crit ≈ . and ˜ m ∞ ≈ . . λ . By analogy with epidemiology we define a dimensionless neurodegenerative reproduction number R = ˜ λ crit λ , (15)such that for R < the protein toxic level is negligible and grows to finite value for R > .The existence of a critical clearance rate shows that in the healthy regime, i.e. for sufficientlylarge values of clearance, the system (1)-(3) with size-independent clearance can support a small,endemic, population of toxic proteins. The aggregation of a significant toxic population, in thiscase, occurs only when the system’s clearance rate, λ , drops sufficiently below the critical clearancerate λ crit . We can explore the dynamics close to the bifurcation by considering the normal form ofthe system for the perfect system ((5)-(7) with k = k n = 0 ) near λ = ˜ λ crit . The general method toobtain the normal form of a transcritical bifurcation for an arbitrary smooth vector field is given inAppendix A. Applying these ideas, we can approximate the full system by ˙ P = − ( λ − ˜ λ crit ) P + αv P P , (16) ˙ M = − ( λ − ˜ λ crit ) M − αM , (17) ˙ m = − ( λ − λ crit )( m − m ) + α ( m − m ) , (18)8igure 4: Toxic mass concentration M ( t ) as a function of time for the unseeded (black dashed)system (5)-(7), for the perfect seeded system (neglecting homogeneous and heterogeneous primarynucleation) (red) and the normal form approximation of M ( t ) (dotted blue). The initial conditionswere selected so that the initial growth rates matched the initial growth rate of the unseeded system.Taking S = 2 . × − to be a small seed value, the red curve was generated with unseededinitial conditions; the black dashed curve was computed using seeded initial conditions given by ( P (0) , M (0) , m (0)) = ( S, S/ , m − S ) ; and, the blue dashed curve was generated by solving (17)with M (0) = S/ . Parameters are for the A β
40 values of Table 1 and λ = 1 / .where α is given by (14) and v P = − ˜ λ crit (cid:16) k + m − k off + ˜ λ crit (cid:17) . (19)Fig. 4 shows a comparison of the total toxic mass evolution, versus time, obtained for the imperfectunseeded system, the perfect seeded system, and the normal form. As expected, the agreement isexcellent as long as the system is close enough to the bifurcation point. Next, we consider the effect of clearance on size distribution. First, we take k off = 0 as suggestedby the data sets. Since, we are interested in the asymptotic size distribution, we can assume that p = m ∞ in Eqs. (1–3), in which case, we have simply that p i = 2 k + m ∞ λ + 2 k + m ∞ p i − = δ p i − , ⇒ p i = δ i − p , i > . (20)Using the definition of M = (cid:80) i> ip i , we obtain: p = M ∞ (1 − δ ) − δ , ⇒ p i = M ∞ δ i − (1 − δ ) − δ i > . (21)9igure 5: The effect of the parameter k off on the size distribution can be appreciated by computing δ/δ as a function of k off . We see that for k off < m ∞ k + , the role of k off is negligible. The dashedcurves are given by the asymptotic approximation (25). Parameters are for the A β
40 values ofTable 1 and λ = 1 / .This analysis is not valid for λ → . In that case, the total mass of the system is systematicallytransferred to larger and larger particles and in the long-time limit all finite aggregate concentrationstend to vanish and the trivial distribution is p i = p i − = p = 0 . However, in that limit, theassumption k off = 0 is not justified anymore as even a small value of k off allows for a non-trivial sizedistribution. Indeed, with k off (cid:54) = 0 , we have the following reccurence relation for c i − λ i p i + 2 k + m ∞ ( p i − − p i ) + 2 k off ( p i +1 − p i ) , i > , (22)with a single bounded solution fo the form p i = δ i − p , i > . (23)with δ = k + m ∞ k off + 12 + λ k off − (cid:115) (2 k off + 2 k + m ∞ + λ ) k off − k + m ∞ k off . (24)An asymptotic expression of δ for small and large values of k off gives: δ = (cid:40) δ (1 − λk off (2 m ∞ k + + λ ) ) + O (cid:0) k off (cid:1) , for k off < m ∞ k + ,δ m ∞ k + + λ k off + O (cid:0) k − off (cid:1) , for k off > m ∞ k + . (25)We see that unless λ = 0 , the role of k off , when sufficiently small, is negligible. We conclude thatclearance (or depolymerization) is sufficient to obtain a non-degenerate size distribution. Next, we assume that clearance of an aggregate depends on its size. In this case, there is no simple,closed equation for the moments, as in Sec. 3, and we must study the full system. Here, we make a10ey assumption about the dependence of the clearance on the aggregate size. We assume that thereexists a critical aggregate size, N , such that all aggregates of size N , or greater, are too large to becleared. Explicitly, this assumption implies that λ i = 0 , ∀ i ≥ N . We also assume that k off = 0 and n c = n = 2 then (1)-(3) can be writtend ˜ M d t = − N − (cid:88) i =2 λ i ip i + 2 k + 2 k n p + 2 k + p P + 2 k σ ( p ) ˜ M , (26)d p d t = λ ( m − p ) − k − k n p − k + p P − k σ ( p ) ˜ M , (27)d p d t = − λ p + k + k n p − k + p p + k σ ( p ) ˜ M , (28)d p i d t = − λ i p i + 2 k + p ( p i − − p i ) . i > , (29)where P = (cid:80) ∞ i =2 p i and ˜ M = (cid:80) ∞ i =2 ip i . The unseeded initial conditions for this system are p (0) = m , p i (0) = 0 for ≤ i, ˜ M (0) = 0 . (30)In general, there is no guarantee of mass conservation. For instance, if λ i ≤ λ ∀ i > and thereis at least one i ≥ such that λ i < λ , then the overall mass of proteins will increase in time asshown in Appendix B. To study the dynamics of (26)-(29), we introduce a finite system with equivalent dynamics. Here,we follow [19] (see also [20]) and introduce a super-particle, denoted q N , which represents theconcentration of all aggregates of size greater than or equal to N : q N = ∞ (cid:88) i = N p i , (31)Since λ i = 0 for all i ≥ N ; we can take the limit of the partial sums of (29) to obtaind q N d t = ∞ (cid:88) i = N k + p ( p i − − p i ) = lim j →∞ j (cid:88) i = N k + p ( p i − − p i ) = 2 k + p p N − − k + lim j →∞ p p j . (32)Since the monomer concentration p , remains bounded, for any fixed time, the last term of (32)tends to zero as j → ∞ and the super particle concentration satisfies the equationd q N d t = 2 k + p p N − . (33)We will distinguish the finite system with a super-particle from the infinite system (26)-(29) byintroducing the notation q i = p i for i < N . Defining Q = (cid:80) Ni =2 q i , and using (33), the corresponding11uper-particle system is defined byd M d t = − N − (cid:88) i =2 λ i iq i + 2 k + 2 k n m + 2 k + mQ + 2 k σ ( m ) M, (34)d m d t = λ ( m − m ) − k − k n m − k + mQ − k σ ( m ) M, (35)d q d t = − λ q + k + k n m − k + mq + k σ ( m ) M, (36)d q i d t = − λ i q i + 2 k + m ( q i − − q i ) , i = 2 , . . . , N − , (37)d q N d t = 2 k + mq N − . (38)The unseeded conditions for (34)-(38) are m (0) = m , q i (0) = 0 , for ≤ i ≤ N, M (0) = 0 . (39)For unseeded initial conditions, the dynamics of the finite system is equivalent to the infinite onein the following sense: First note that ˙ Q = ˙ P ; this follows directly from the definition of Q , P and(31). Thus, Q and P will agree, for all time. In turn, (26) and (34) coincide when the initial data(30) and (39), respectively, are used; thus ˜ M ( t ) = M ( t ) in this case. Finally, by definition, p i = q i for ≤ i < N and (31)-(32) has already established that solving (38) produces q N ( t ) = (cid:80) ∞ i = N p i ( t ) provided the initial conditions agree. The above establishes an important fact that we rely on forthe rest of the section; solving (26)-(29) with initial conditions (30) and solving (34)-(38) with initialconditions (39) yields m ( t ) = p ( t ) , Q ( t ) = P ( t ) , M ( t ) = ˜ M ( t ) , (40) p i ( t ) = q i ( t ) for ≤ i < N and q N ( t ) = ∞ (cid:88) i =1 p i ( t ) . We remark, however, that M ( t ) , defined as the solution of (34), is the total toxic mass of both (26)-(29) and (34)-(38), due to (40), for the unseeded initial conditions (39); however, M ( t ) cannot beconstructed a posteriori from the knowledge of q i ( t ) where i = 2 , , . . . , N in the same manner that ˜ M ( t ) can be retrieved from the knowledge of the p i ( t ) . That is, we have M ( t ) (cid:54) = (cid:80) Ni =2 iq i ( t ) . Indeed,in the closure process of reducing the full system to a finite one, we lost information regarding themass of individual particles making up the superparticle. Nevertheless, both the evolution of toxicmass of the full system, as well as the size distribution (up to size N ) can be obtained by studyingthe finite system (34)-(38). Systems such as (26)-(29) or (34)-(38), with size-dependent clearances, do not conserve mass ingeneral (see Appendix B) and the toxic mass may increase with time. We study in more details theparticular choice λ i = λ/i, for i = 1 , , . . . , N − , (41)12hich expresses the modeling assumption that aggregates become increasingly difficult to clearas their size increases. An example of the dynamics of the system (34)-(38) is shown in Fig. 6.We observe two different behaviors. Initially, up to a time τ , the system mostly behaves likethe conservative no-clearance model ( λ = 0 ) even for large values of clearance. This behavior ismarkedly different than the one observed in Fig. 2. Second for larger times, t > τ , the monomermass always decreases and the toxic mass always increases as predicted from our general analysis.We observe that larger clearance leads to faster toxic mass creation. This is due to the fact that inhealthy homeostasis, production and clearance are balanced. Hence larger clearance implies largerproduction. The question is then to understand the transition between the two regimes as well asthe small and large time behaviors of all species.Figure 6: Toxic mass dynamics for the size-dependent clearance λ i = λ/i ; the monomer populationconcentration ( m ( t ) , blue lines) and total toxic mass ( M ( t ) , red lines) are shown for clearance (inh − ) rates: λ = 0 (dashed), λ = 0 . (solid), and λ = 1 (dotted). Parameters are for the A β λ = 1 / N = 20 . On long time scales, i.e. long enough so that the monomer concentration begins to decrease, themonomer production, aggregation, and nucleation processes result in an increase to subsequent toxicspecies and, therefore, to the overall toxic mass M . The asymptotic behavior of the system toxicmass M is observed to depend entirely on the production rate, γ = λm , as M ( t ) ∼ t →∞ γ t. (42)13igure 7: Long time (in h ) concentration (in moles) dynamics of (34)-(38) ( λ i = λ/i , in h − ) with N = 20 and A β
40 parameters (Table 1, third column); curves for λ = 0 . (solid) with asymptoticslopes (dotted). Time runs from 0 to 5000 hours. Parameters are for the A β
40 values of Table 1and N = 20 .This behavior is illustrated in a log-plot in Fig. 7; the characteristic time scale, τ , indicates thetime at which the monomer mass begins to decay. Once the asymptotic behavior of M has beenestablished, the equations can be balanced asymptotically by the following dynamics: m ∼ α m t − / , q N ∼ α N t / , q i ∼ α i t / , i = 1 , . . . , N − , (43)where the symbol “ ∼ ” is understood as the long-time asymptotic behavior and the α i are constants.This asymptotic behavior shows that the super-particle dominates the long-term dynamics; thus P ∼ q N for large times. Physically, in the long-time limit, the monomer population, renewed bythe continuous production, is quickly promoted to the super-particle through linear aggregation. We observe in Fig. 6 that the early-time behavior is not greatly perturbed by altering the clearancerate. Hence, we can obtain characteristic time scales for the amplification of the toxic mass byconsidering the limit λ → + . In this case, the early evolution of the toxic mass is governed by thedynamics of (5)-(7) with λ = 0 . There are two characteristic time scales of importance. First, thetime scale τ associated with the exponential growth of the toxic mass in early time via the inverseof the positive linear eigenvalue, µ = 1 /τ , corresponding to the linearization of (5)-(7) around thehealthy state m = m , M = P = 0 . The linear eigenvalue is given by the positive root of µ + µ (cid:18) m k n − k m K M K M + m (cid:19) − k + k m K M K M + m + 4 k + m k n = 0 . (44)14econd, there is a time scale τ where both nucleation and amplification are balanced. It is givenby the time for the linearized solution for M ( t ) to reach m . Hence τ is the solution of m = (cid:0) m k n + k (cid:1) (cid:0) K M + m (cid:1) k n K M + 2 m k n − k m K M (cid:32) − e τ /τ (cid:33) . (45)For example, for the first parameter set (A β
40) used for the figures, these times are τ ≈ . h and τ ≈ . h. The value of τ is a rudimentary estimate for the time of amplification; it is a lowerbound for the typical time scale of growth (see Fig. 6). Nevertheless, in Fig. 7, we see that τ canindeed act as an indicator for the onset of decay for the monomer mass. A more refined estimatecan be obtained by using the approximate solution for the full dynamics given in [12]. Another interesting case to consider is when the population of monomer is not depleted but remainsat a constant level m . We assume that, regardless of other parameters, that (1) is instead specifiedby d p d t = 0 , (46)so that, with unseeded initial conditions, we have p ( t ) = m for all time. Assuming again nodepolymerization, no fragmentation, and dimer nucleation, the master equations now readd p d t = − λ p + k + k n m − k + m p + k σ M, (47)d p i d t = − λ i p i + 2 k + m ( p i − − p i ) , i > , (48)where σ = σ ( m ) and M = (cid:80) ∞ i =2 ip i is the total toxic mass. This is an infinite system of linearordinary differential equations. For this system, we consider three types of clearance; the size-independent case in addition to two different size-dependent paradigms. All three clearance relationscan be summarily presented by a power-law of the form λ i = λi ν . (49)When ν = 0 we recover the size-independent case; when ν = − we recover the size-dependentdiminishing clearance formulation used in Sec. 4; and, finally, the case of ν = 1 corresponds toimproved clearance, with increasing size, which could arise due to, for instance, antibody binding.Depending on the two parameters λ and ν , the solution to this system may have a steady stateor increase indefinitely. The question is then to identify the critical values at which this transitionhappens. We start with the simple case of constant clearance ν = 0 ; this is the analogue to Sec. 3 for aconstant free monomer assumption (c.f. (46)) The moments (c.f. Sec 3) are specified by a simple15air of linear equations given byd P d t = − λP + k + k n m + k σ M, (50)d M d t = − λM + 2 k + 2 k + mP + 2 k n m + 2 k σ M, (51)which can be written as ˙ q = A q + b , (52)where q = ( P, M ) T , b = ( k + k n m , k + 2 k n m ) T and A = (cid:18) − λ ab a − λ (cid:19) = (cid:18) − λ k σ k + m k σ − λ (cid:19) . (53)The constant solution sole steady state for this system is q ∞ = − A − b ; q ∞ is positive and finite if λ > a + (cid:112) a + ab = k σ + (cid:113) k σ + 2 k σ k + m = λ (0) crit . (54)This condition naturally provides a value for the critical clearance. Specifically, the largest lineareigenvalue for the system is κ = λ (0) crit − λ ; solutions converge to q ∞ exponentially in time (as e κt ) for λ > λ (0) crit and grow unbounded for λ ≤ λ (0) crit . The values given in Table 1 for the different parametersshow that this estimate is indistinguishable from the case studied in Section 3, which is explainedby the fact that at the bifurcation point, the monomer population is constant in both cases. We now turn our attention to the general case where the clearance terms are not constant. Then, themaster equations do not yield a closed system for the moments. Nevertheless, due to the simplicityintroduced by p ( t ) = m being constant, we can find conditions for the existence of a fixed-pointsolution, ( p ∗ , p ∗ , . . . ) to (47)-(48). If such a steady state p ∗ i for i > , exists, it must satisfy therecurrence relation p ∗ i = δ i p ∗ i − , δ i = bb + λ i = 2 k + m k + m + λ i . (55)we note that each of the recursion coefficients, δ i , is now dependent on i via λ i . Define a sequenceof real numbers, indexed by i , as ∆ i = i (cid:89) j =3 δ j , i > . (56)We define ∆ = 1 and the i th steady state is expressible, for all i ≥ , through its recurrence relationas p ∗ i = ∆ i p ∗ , i ≥ . (57)Defining ∆ = ∞ (cid:88) j =3 ∆ j , (58)16he steady state for the total toxic mass solution M ∗ is then given by M ∗ = ∞ (cid:88) i =2 i ∆ i p ∗ = ∆ p ∗ , (59)and an application of (47), at steady state, gives the value of p ∗ as p ∗ = k + k n m λ + 2 k + m − k σ ∆ . (60)Therefore, for a fixed point to exist we need the three following conditions to be satisfiedC1: lim i →∞ ∆ i = 0 , (61)C2: ∆ = ∞ (cid:88) i =2 i ∆ i converges , (62)C3: k σ ∆ − λ − k + m > . (63)An analysis of the case ν = 0 recovers the previous condition and it can then be verified directlythat conditions C1-C3 are satisfied, as expected, for λ > λ (0) crit . ν = 1 For ν = 1 , we have (see Appendix C), ∆ (1) = 2 + b/λ and the steady population of dimers, wheneverit exists, is given by p ∗ = λ ( k − k n m )2( k + m + λ )( λ − k σ ) . (64)Hence, condition C3 leads to λ > λ (1) crit with λ (1) crit = k σ . (65)We note that the above implies that the critical clearance depends only on the secondary nucleationprocess and, in particular, not the process of elongation (c.f. λ (0) crit in (54)). ν = − For ν = − , the situation is not as simple. The condition C1 is verified but C2 leads to λ > b forwhich ∆ ( − = ( λ + 2 b ) (cid:16) Γ (cid:0) λb − (cid:1) Γ (cid:0) λb + 2 (cid:1) − Γ (cid:0) λb (cid:1) (cid:17) b Γ (cid:0) λb (cid:1) , (66)where Γ( · ) is the usual Gamma function. Condition C3 is satisfied if λ > λ ( − crit where λ ( − crit is thepositive solution of f ( λb ) = 1 + 2 k + m k σ , with f ( λb ) = Γ (cid:0) λb − (cid:1) Γ (cid:0) λb + 2 (cid:1) Γ (cid:0) λb (cid:1) . (67)17his equation always has a solution as f : z ∈ [2 , ∞ ] → f ( z ) is such that f (cid:48) ( z ) < , f ( z ) → z → ∞ and f ( z ) → z →∞ . For the parameters listed in Table 1, k + m /k σ (cid:29) , in which case, we canapproximate the function f ( z ) close to z = 2 by f ( z ) ≈ / ( z − , which leads to the critical value λ ( − crit = 8 k + m (2 k σ + k + m ) k σ + 2 k + m (68)This last relation can be further simplified by realizing that k + m (cid:29) k σ , which leads to λ ( − crit = 4 k + m . (69)For the parameters given in Table 1, this last approximation of the critical clearance gives thecorrect value (compared to (67)) to 6 digits. Note that, in contrast to the critical clearance ratefor enhanced clearance (c.f. (65)), (68) depends only on the elongation rate k + . In particular, in areduced clearance regime, a change in the rate of secondary nucleation has no effect on the clearancerate required to keep the system stable. The general trend that can be observed from Table 1 isthat λ ( − crit > λ (0) crit > λ (1) crit , as expected. ν = − Finally, for ν = − , skipping computational details, we find that lim n →∞ ∆ ( − n = 14 π (cid:114) λb (cid:32)(cid:18) λb (cid:19) + 5 λb + 4 (cid:33) csch (cid:32) π (cid:114) λb (cid:33) , (70)which is positive for all finite positive value of λ . Hence, condition C1 is not satisfied and there isno constant solution or critical value of the clearance that would limit unbounded growth of toxicproteins. We note that we have neglected the effect of fragmentation. For ν < , the effect offragmention is the creation of smaller aggregates that increase the overall expansion of the proteinpopulation but also boosts clearance. Indeed since smaller aggregates are more likely to be clearedand we expect a reduction of the critical value of clearance as well as the possibility of a finite valueof clearance for ν = − or smaller as shown in Meisl [27]. Comparing the different critical clearancevalues given in Table 1 for the three values of ν , it is clear that that the choice of clearance lawhas a significant impact on the clearance values as they differ, from the smallest to the largest by9 orders of magnitude. Hence, enhancing or inhibiting the clearance mechanism may be extremelyimportant to the overall increase of toxic proteins. We have assessed the impacts of production and clearance on the aggregation kinetics using atheoretical model, c.f. (1)-(3), that has been experimentally validated [8, 15, 16]. Our findingssuggest that clearance may mediate toxic aggregation kinetics. In the case of constant clearance,we showedthat toxic aggregation is controlled, directly, by a critical clearance. Clearance above thislevel provides for a robust environment which is, essentially, free of toxic proteins; clearance belowthis level triggers and instability and a propensity towards toxic mass accumulation. Once toxicaggregation is triggered, the healthy monomer population is diminished as aggregates form. The18aximal amount of toxic formation is, again, mediated by the clearance level; an effect of the massconservation principle, of this regime.A reasonable in vivo hypothesis is that the clearance may depend on the aggregates size i .This clearance paradigm has been explored using a simple inverse proportionality law λ i = λ/i .The resulting set of equations, for this type of clearance, does not yield a finite system for themoments; thus, a super-particle system, with identical trajectories in the presence of unseeded initialconditions, has been advanced as a means of study. In the presence of any aggregation effects,the system immediately begins accumulating toxic mass; even from unseeded initial conditions.Moreover, mass is not conserved and the toxic mass grows unboundedly in time. The clearance,however, determines the asymptotic rate of increase of the toxic mass as a function of time with M ( t ) ∼ λm t . The biological implications of a size-dependent clearance are quite different than theconstant case. In particular, if clearance is size-dependent, results suggest that we have no recoursein halting aggregate pathology through enhancing clearance; rather, we can only hope to delay theoverall trend of toxic accumulation.The theoretical model of a constant free-monomer concentration was also consdired. This case isparticularly interesting since, under the assumption of steady states, we see that a notion of criticalclearance can be established for relations of the form λ i = λi ν for ν ∈ {− , , } . In the case of ν = 0 , we recover the previous results given for constant clearance. Similarly, in the size-dependentcase ( ν = − and ν = 1 ), there exists a critical value of the clearance so that no aggregationtakes place past that value. Remarkably, our results suggest that, depending on the specific size-dependence, the processes of elongation and secondary nucleation contribute to the value of thecritical clearance to different degrees. An important implication is that, depending on the specificmechanism of clearance, inhibition of aggregation should target different processes in order to reducethe critical clearance rate.Overall, the role of clearance in aggregation kinetics is highly non-trivial. However, our studyshows that clearance may play an important role in the aggregation kinetics of Amyloid- β and thatadditional experiments, providing fitted values for clearance parameters, would serve to elucidateappropriate regimes for further study. Acknowledgments–
This work was supported by the Engineering and Physical Sciences Re-search Council grant EP/R020205/1 to Alain Goriely and by the John Fell Oxford University PressResearch Fund grant 000872 (project code BKD00160) to Travis Thompson.
A Normal form for a transcritical bifurcation
Here we derive the normal form of a transcritical for a general dynamical system. We consider anautonomous n -dimensional C vector field of the form ˙ x = f ( x , λ ) , x ∈ R n , (71)and assume that there exists a constant solution x such that f ( x , λ ) = and a different equilibriumsolution in a neighborhood of the critical value λ . The conditions for the existence of a transcriticalbifurcation at the critical value λ are given by Sotomayor’s theorem [21] and the reduced form thesystem takes close to that value can be captured by normal form theory [22, 23, 24, 25]. Here, weuse multiple scale analysis to obtain a convenient form of the reduced equations. The result in itself19s not original but it may not be obvious to find a direct reference for either the statement or theproof. Therefore, its inclusion may be helpful to the reader.Using multiple-scale expansion, we expand the solution as x = x + (cid:15) x + (cid:15) x + . . . , λ = λ + (cid:15)λ . (72)where x is constant and x i = x i ( t, τ ) , i>1 and τ = (cid:15)t is a slow time [26]. The expansion of thevector field close to second order is f = f (73) + (cid:15) [ D f · x + f λ, λ ] (74) + (cid:15) (cid:20) D f · x + 12 H f ( x , x ) + λ D f λ, · · · x (cid:21) (75) + . . . , (76)where f = f ( x , λ ) indicates that f is evaluated at the point ( x , λ ) and ( D f ) ij = ∂f i ∂x j , D f = D f ( x , λ ) , (77) f λ = ∂ f ∂λ , f λ, = f λ ( x , λ ) , (78) ( H f ) ijk = ∂ f i ∂x j ∂x k , H f = H f ( x , λ ) , (79) ( D f λ ) ij = ∂ f i ∂x j ∂λ , D f λ, = D f λ ( x , λ ) . (80)If the system has a bifurcation of co-dimension one at λ then D f has rank n − and the followingvectors w and v given by w · D f = , D f · v = , (81)define the left and right null spaces of D f . The generic condition for a transcritical bifurcation tooccur is w · f λ, = 0 . (82)To order O ( (cid:15) ) , the differential equation reads ˙ x = D f · x + λ f λ, . (83)and we are interested in the solution x = c ( τ ) v , (84)whose existence is guaranteed by the condition w · f λ, = 0 . To second order O ( (cid:15) ) , we have ˙ x + c (cid:48) ( τ ) v = D f · x + c H f ( v , v ) + cλ D f λ, · v . (85)The Fredholm alternative gives a condition for the existence of a solution of this inhomogeneoussystem: w · ( c (cid:48) ( τ ) v ) = w · (cid:18) c H f ( v , v ) + cλ D f λ, · v (cid:19) , (86)20hich gives the equation c (cid:48) ( τ ) = βλ c + αc , (87)where α = 12 1 v · w w · H f ( v , v ) (88) β = 1 v · w w · D f λ, · v . (89)Taking into account that (cid:15)λ = λ − λ and defining y = (cid:15)c , the local solution is x = x + y v where ˙ y = β ( λ − λ ) y + αy , (90)is the normal form of a transcritical bifurcation at λ = λ . The local evolution of the variables forwhich v i (cid:54) = 0 is given by ˙ x i = β ( λ − λ )( x i − x ,i ) + αv i ( x i − x ,i ) . (91) B Mass balance in the size-dependent clearance case
For unseeded initial conditions, we can show that the total mass of the system is not conserved.Assume that, for all ≤ i we have λ i ≤ λ and assume that there exists some index j , with ≤ j ,such that the inequality is strict (i.e. λ j < λ ). In this case we haved ˜ M d t > − λ N − (cid:88) i =2 ip i + 2 k + 2 k n p + 2 k + p P + 2 k σ ( p ) ˜ M> − λ ∞ (cid:88) i =2 ip i + 2 k + 2 k n p + 2 k + p P + 2 k σ ( p ) ˜ M = − λ ˜ M + 2 k + 2 k n p + 2 k + p P + 2 k σ ( p ) ˜ M . (92)Likewise for i = 2 we have a similar inequalityd p d t = − λ p + k + k n p − k + p p + k σ ( p ) ˜ M ,> − λ p + k + k n p − k + p p + k σ ( p ) ˜ M , (93)and likewise for i > . The above observation shows that the system (26)-(29) grows faster than theconstant-clearance case system where λ i = λ for every i ∈ { , , . . . } . We note that, as in Sec. 3,the total system mass for (26)-(29) is ˜ M tot = ˜ M + p ; this follows from the common definition of ˜ M , here, and M (see (4)). Adding (26) to (27) and using (92) givesd ˜ M tot d t > λ (cid:16) m − ˜ M tot (cid:17) . (94)In the presence of the unseeded initial conditions (30) we have that ˜ M tot (0) = m so that theleft-hand side of (94) is strictly positive and mass conservation is violated at the outset. Now let21 λ tot denote the total mass of the constant clearance case λ i = λ for all i . We know that, in thepresence of unseeded initial conditions, a sysetm with constant clearance systems conserves massso that d M λ tot d t = 0 . From (92) and (93), which holds analagously for i > and for i = 1 we have equality, we canconclude that d ˜ M tot d t ≥ d M λ tot d t = 0 , (95)for unseeded initial conditions. Take together, (94) implies that the system (26)-(29), with unseededinitial conditions, initially gains mass while (95) shows that it can never lose mass. Therefore, notonly does (26)-(29) not conserve mass but it can never return to the state of initial unseeded mass. C Critical value for enhanced clearance
For ν = 1 , the case (56) takes the form ∆ i = i (cid:89) m =3 (cid:18) bb + mλ (cid:19) = ( b + λ )( b + 2 λ ) b (cid:18) bλ (cid:19) i (cid:18)(cid:18) b + λλ (cid:19) i (cid:19) − , i ≥ , (96)where the subscript ( x ) i = x ( x + 1)( x + 2) · · · ( x + i − denotes ascending factorial (i.e. thePochhammer symbol). Defining ξ = bλ − then (C1) is satisfied provided lim i →∞ ξ i ( ξ + 1) i = 0 . (97)The function ξ i (( ξ + 1) i ) − is monotonically decreasing in both ξ and i and condition C1 is satisfiedfor any ξ > . Using ξ = bλ − the expression (96) implies ∆ (1) = 2 + ( λ + λξ )(2 λ + λξ ) λ ξ ∞ (cid:88) i =3 i ξ i ( ξ + 1) i = 2 + ( λ + λξ )(2 λ + λξ ) λ ξ (cid:18) ξ ξ + ξ (cid:19) = 2 + ξ. Thus we have ∆ (1) = 2 + bλ , (98)and it follows that p ∗ , for ν = 1 , is determined by the formula p ∗ = λ ( k − k n m )2( k + m + λ )( λ − k σ ) . (99)22 eferences [1] John A Hardy and Gerald A Higgins. Alzheimer’s disease: the amyloid cascade hypothesis. Science , 256(5054):184–186, 1992.[2] John Hardy and David Allsop. Amyloid deposition as the central event in the aetiology ofalzheimer’s disease.
Trends in pharmacological sciences , 12:383–388, 1991.[3] Dennis J Selkoe and John Hardy. The amyloid hypothesis of Alzheimer’s disease at 25 years.
EMBO molecular medicine , 8(6):595–608, 2016.[4] Andrew Bacyinski, Maosheng Xu, Wei Wang, and Jiani Hu. The paravascular pathway forbrain waste clearance: Current understanding, significance and controversy.
Front. Neuroanat. ,11:101, 2017.[5] H Benveniste, X Liu, S Koundal, S Sanggaard, H Lee, and J Wardlaw. The glymphatic systemand waste clearance with brain aging: A review.
Gerontology , 65:106–119, 2019.[6] Jenna Tarasoff-Conway, Roxana Carare, and Mony J. et. al. de Leon. Clearance systems in thebrain–implications for Alzheimer disease.
Nat Rev Neurol , 11(8):457–470, 2015.[7] Shu-Hui Xin, Lin Tan, Xipeng Cao, Jin-Tai Yu, and Lan Tan. Clearance of Amyloid Beta andTau in Alzheimer’s Disease: from Mechanisms to Therapy.
Neurotoxicity Research , 34(3):733–748, 2018.[8] G. Meisl, X. Yang, E. Hellstrand, B. Frohm, J. Kirkegaard, S. Cohen, C. Dobson, S. Linse, andT. Knowles. Differences in nucleation behavior underlie the contrasting aggregation kinetics ofthe aA β
40 and A β
42 peptides.
Proceedings of the National Academy of Sciences , 111(26):9384–9389, 2014.[9] Samuel IA Cohen, Michele Vendruscolo, Mark E Welland, Christopher M Dobson, Eugene MTerentjev, and Tuomas PJ Knowles. Nucleated polymerization with secondary pathways. I.time evolution of the principal moments.
The Journal of chemical physics , 135(6):08B615,2011.[10] Samuel IA Cohen, Michele Vendruscolo, Christopher M Dobson, and Tuomas PJ Knowles. Nu-cleated polymerization with secondary pathways. II. determination of self-consistent solutionsto growth processes described by non-linear master equations.
The Journal of chemical physics ,135(6):08B611, 2011.[11] Samuel IA Cohen, Michele Vendruscolo, Christopher M Dobson, and Tuomas PJ Knowles.Nucleated polymerization with secondary pathways. III. equilibrium behavior and oligomerpopulations.
The Journal of chemical physics , 135(6):08B612, 2011.[12] G. Meisl, J. Kirkegaard, P. Arosio, T. Michaels, M. Vendruscolo, C. Dobson, S. Linse, andT. Knowles. Molecular mechanisms of protein aggregation from global fitting of kinetic models.
Nature protocols , 11(2):252, 2016. 2313] Rebecca Frankel, Mattias Törnquist, Georg Meisl, Oskar Hansson, Ulf Andreasson, HenrikZetterberg, Kaj Blennow, Birgitta Frohm, Tommy Cedervall, Tuomas PJ Knowles, et al. Au-tocatalytic amplification of Alzheimer-associated A β
42 peptide aggregation in human cere-brospinal fluid.
Communications biology , 2(1):1–11, 2019.[14] Franziska Kundel, Liu Hong, Benjamin Falcon, William A McEwan, Thomas CT Michaels,Georg Meisl, Noemi Esteras, Andrey Y Abramov, Tuomas JP Knowles, Michel Goedert, et al.Measurement of tau filament fragmentation provides insights into prion-like spreading.
ACSchemical neuroscience , 9(6):1276–1282, 2018.[15] Samuel IA Cohen, Sara Linse, Leila M Luheshi, Erik Hellstrand, Duncan A White, LukeRajah, Daniel E Otzen, Michele Vendruscolo, Christopher M Dobson, and Tuomas PJ Knowles.Proliferation of amyloid- β
42 aggregates occurs through a secondary nucleation mechanism.
Proceedings of the National Academy of Sciences , 110(24):9758–9763, 2013.[16] S. Linse, T. Scheidt, K. Bernfur, M. Vendruscolo, C. Dobson, S. Cohen, E. Sileikis,M. Lundquist, F. Qian, T. O’Malley, et al. Kinetic fingerprint of antibody therapies predictsoutcomes of Alzheimer clinical trials. bioRxiv , page 815308, 2019.[17] D.L. Brody, H. Jiang, and N. et al. Wildburger. Non-canonical soluble amyloid-beta aggregatesand plaque buffering: controversies and future directions for target discovery in Alzheimer’sdisease.
Alz. Res. Therapy , 9, 2017.[18] Formari S., Schäfer A., Goriely A., and Kuhl E. Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading inthe brain and its connectome.
J. Theor. Biol. , 2019.[19] M. Bertsch, B. Franchi, N. Marcello, M. C. Tesi, and A. Tosin. Alzheimer’s disease: a math-ematical model for onset and progression.
Mathematical Medicine and Biology , page dqw003,2016.[20] S. Fornari, A. Schäfer, E. Kuhl, and A. Goriely. Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading inthe brain and its connectome.
Journal of Theoretical Biology , 486:110102, 2020.[21] J. Sotomayor. Generic bifurcations of dynamical systems. In
Dynamical systems , pages 561–582. Elsevier, 1973.[22] J. Guckenheimer and P. Holmes.
Nonlinear oscillations, dynamical systems and bifurcations ofvector fields . Springer-Verlag, New York, 1983.[23] S. Wiggins.
Global bifurcations and chaos . Springer-Verlag, New York Berlin, 1988.[24] A. Goriely.
Integrability and Nonintegrability of Dynamical Systems . World Scientific PublishingCompany, 2001.[25] A. Goriely. Painlevé analysis and normal forms theory.
Phys. D , 152:124–144, 2001.[26] A. Newell. Envelope equations.
Lect. Appl. Math. , 15, 1974.2427] G. Meisl. Modelling protein aggregation in-vitro and in-vivo.