The role of clearance mechanisms in the kinetics of pathological protein aggregation involved in neurodegenerative diseases

Alzheimer’s disease (AD) and other related neurodegenerative diseases are associated with the assembly of specific proteins into pathological fibrillar aggregates. Alzheimer’s disease, in particular, is characterized by the aggregation of amyloid-β (Aβ) into plaques and of the tau protein into neurofibrillary tangles (NFTs). The role of Aβ in Alzheimer’s is thought to be so central to the disease that it is the basis of the so-called “amyloid-β hypothesis,”^1–3 stating that the accumulation and deposition of oligomeric or fibrillar amyloid-β peptide are the main cause of the disease. This hypothesis has provided a guide for most of the AD research over the last 20 years. However, recent experimental evidence, and the failure of several drug trials, has led to renewed scrutiny of this foundational assumption, and next generation therapeutic intervention strategies that target more specifically the low molecular weight oligomers are showing more promise.⁴ 

The production of Aβ is a natural process related to neuronal activity. Indeed, Aβ is a normal metabolic waste by-product^5,6 that is typically removed from intracellular and extracellular compartments by several clearance mechanisms.^7,8 In healthy subjects, waste proteins are broken down by enzymes, removed by cellular uptake, crossing the blood–brain barrier, or efflux to cerebrospinal fluid compartments where they eventually reach arachnoid granulations, or lymphatic vessels. While healthy clearance mechanisms, working in harmony, avert the buildup of toxic Aβ plaques and tau NFT, their impairment or dysfunction can lead to pathology.⁸ The specifics of in vivo clearance mechanisms remain a topic of clinical debate; however, the kinetics enabling proteins to amass into pathological aggregates can be carefully, and systematically, studied in vitro and under varied conditions. The production of Aβ, at a high level, is mediated by a membrane protein, the amyloid precursor protein (APP). APP is typically cleaved by α-secretase, and the resulting products do not aggregate. However, APP can also be cleaved by β-secretase and γ-secretase, a process that results in soluble monomeric APP fragments of different sizes. The most common variants are Aβ38, Aβ40, and Aβ42. While monomeric Aβ38 is not prone to further aggregation, Aβ40 and Aβ42, containing additional amino 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.⁹ In fact, it has recently been shown¹⁰ that the aggregation properties of Aβ40, which is more abundant, differ from those of the more aggregate-prone Aβ42 even under the same conditions. A theoretical framework of chemical kinetics and aggregation theory^11–13 has been combined with careful, systematic in vitro experiments performed under differing conditions, such as varied concentration or pH. This approach based on chemical kinetics has elucidated effective pathways and mechanisms for nucleation, aggregation, and fragmentation¹⁴ and produced a deep understanding of key properties, underlying the formation of aggregates under ideal conditions, with the potential for therapeutic intervention.^15,16

Here, we extend this framework and develop a mathematical approach to describe the effects of clearance and monomer production on the kinetics of aggregation. We apply the framework to the study of Aβ. To accomplish this objective, we extend the current theory describing Aβ aggregation in vitro, which has been validated against experiment, to include monomer production and oligomer clearance terms. In particular, we study two different clearance mechanisms: one where all aggregated species are cleared at the same rate (size-independent clearance) and the other where different aggregated species may be cleared at different speeds (size-dependent clearance). In the former case, we show the full system reduces to three equations amenable to a systematic analysis. We identify a critical value of clearance above which the production of aggregates does not take place. Our results offer further evidence in support of two main hypotheses that clearance mechanisms play a crucial role in neurodegenerative disease initiation and progression and that therapies enhancing clearance above a prescribed, critical value may serve as a possible intervention strategy. In particular, we will exhibit the existence of critical clearance values, which are consistent with the observation of disease onset when natural clearance mechanisms within the brain have degraded through aging.

Our model for protein aggregation-dynamics includes the key molecular steps: heterogeneous primary nucleation, homogeneous primary nucleation, secondary nucleation, linear elongation, and clearance (cf. Fig. 1). These mechanisms lead to a general class of mathematical models that can describe a wide range of aggregating systems in vitro. In particular, by including heterogeneous primary nucleation terms, a source term for new nuclei, which can become independent of monomer concentration, is present;¹⁷ we include this here in addition to the usual monomer-dependent homogeneous primary nucleation. Thus, in such a model, the importance of interfaces in the initiation of nucleation is sufficiently accounted for. In the model, each aggregate of a given size is represented by a population. In general, each population, with aggregates of size i, will be represented by an indexed concentration. We use the special notation m(t) for the monomer population i = 1, while all other aggregate concentrations are denoted by p_i(t) for $i∈2,3,…$⁠. The master equations are then

where we require i ≥ n_c > 1 and δ_i,j is Kronecker’s delta (δ_i,j = 1 if i = j and 0 otherwise) and

Here, P and M are the first two moments of the population distribution. They represent the total number and total mass concentrations of aggregates, respectively. The functions σ(m) and ρ(m) account for the ability of both secondary nucleation and heterogeneous primary nucleation to saturate. As fibrils can generally grow from both ends, the conventional definition of the rates is with respect to the concentration of fibril ends, which is twice the number concentration of aggregates, giving rise to a factor of 2, in (1) and (2). In these equations, the parameters represent the following effects, sketched in Fig. 1: γ: (constant) monomer production such as by secretase mitigated cleavage of APP, driving mass influx; λ_i: clearance of aggregates of size i via, e.g., cerebrospinal-fluid mediated, blood–brain barrier mediated, or cellular degradation processes; k_h: heterogeneous primary nucleation rate constant; k_n: homogeneous primary nucleation of aggregates; n_c: size of the smallest stable fibril (the nucleus) formed by either process of primary nucleation (n_c > 1); k₊: linear elongation transforming aggregate from size i to i + 1; k₂: secondary nucleation of aggregates of size n₂ ≥ n_c; K_M: saturation constant of the secondary nucleation; K_P: saturation constant of the heterogeneous primary nucleation; and k_off: depolymerization by loss of one monomer from the fibril end. When fitting experimental data, it is often the case that only one of heterogeneous primary nucleation or homogeneous primary nucleation is required to explain the data. We include both here for completeness. Finally, a general note on the choice of model is that the aim here is to provide a tractable model with a focus on the kind of behavior that emerges when aggregate removal processes are active. We have therefore chosen a level of coarse-graining that is sufficient to describe the majority of currently available experimental data. In particular, we use one rate constant per process to describe elongation, de-polymerization, and nucleation. In reality, an aggregated system is likely to be heterogeneous and there may be slight variations of the rate constants, for example, with fibril length.¹⁸ Nevertheless, our model of aggregation in vitro captures the key physics as evidenced by the its successful application to modeling numerous experimental systems well to within the measurement accuracy.^14,19 However, one should keep in mind that the rate constants extracted from experimental analysis necessarily constitute ensemble-averaged quantities, into which such small heterogeneities are subsumed.

The aggregation model [(1) and (2)] and its many variations have been used to describe the results of many in vitro experiments.^10,15,20 The analyses of multiple experimental datasets have shown that the exponents n_c = n₂ = 2 for Aβ40 and for Aβ42 and that these processes show little sign of saturation at low μM concentrations of protein when aggregating in phosphate-buffered saline (PBS) solution.^10,20 In contrast, for Aβ42 aggregating in HEPES buffer, primary nucleation appears to be fully saturated and its rate shows no monomer dependence.¹⁵ While n₂ = 2 still holds under those conditions, primary nucleation appears to proceed purely via heterogeneous primary nucleation that is fully saturated, i.e., K_P ≪ m. In this limit, k_hρ(m) → k_hK_P. For the discussions and derivations in this manuscript, we take the view of Aβ experiments^10,20 so that n_c = n₂ = 2 and that heterogeneous primary nucleation, if it is present, is fully saturated, i.e., k_hρ(m) → k_hK_P = k₀, where we have defined k₀, the rate of heterogeneous primary nucleation in the fully saturated regime, for simplicity. Adaptation to other aggregating systems is straightforward, and all numerical results are qualitatively similar.

The primary purpose of this manuscript is to describe the qualitative impact of clearance mechanisms on the dynamics of protein aggregation. Because primary nucleation is generally not a dominant process in the aggregation of disease associated proteins, the particular choice of nucleation mechanism, such as heterogeneous vs homogeneous, does not affect the results. Examples of fitted Aβ model parameters are listed in Table I. PBS and HEPES refer to the buffers used in the corresponding experiments. In these experiments, aggregation proceeds much faster than depolymerization and k_off = 0 is found to be a good fit to describe the dynamics. However, from a theoretical point of view, we note that k_off = 0 violates detailed balance and implies that 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 addition of this small term does not change our results. Therefore, we will use the fitted experimental parameters given in Table I. Clearance and production have not been investigated experimentally; thus, we leave them as free parameters. Notably, a primary contribution of the current work is in determining particular values of these parameters when a qualitative change of the dynamics occurs.

One key fact to consider before we embark on solving the dynamics is what boundary conditions, in the form of protein homeostasis, we impose on the system. In the aggregation of purified protein in vitro, there are no sources or sinks for protein mass, so the total mass of protein is conserved. In the present case, a living system, both sources and sinks exist and different possibilities arise. Generally, organisms will strive to maintain stable conditions and balance these sources and sinks. We consider two extreme cases and show that our conclusions hold in both regimes and are thus likely to apply in a range of biological systems. First, we will consider a system in which the monomer production rate is constant over time (see Secs. III and IV). Second, we will consider a system where instead the monomer concentration is constant in time, so the monomer production rate may vary (see Sec. V). Most biological systems are likely to lie somewhere between these two extremes.

In the case of size-independent clearance, we have λ_i = λ > 0 for all i. Our main question is to understand the role of the clearance term. In particular, we will establish that if clearance is sufficiently large, the formation of aggregates does not take place. We will then estimate this critical clearance value, denoted λ_crit, under the different experimental conditions considered.

In the size-independent case, a previously established¹¹ but remarkable feature of the system (1) and (2) is that a closed system of equations for the first two moments P and M and the monomer concentration m can be obtained as

where σ(m) = m²K_M/(K_M + m²), and we have chosen n₂ = 2. The total mass of the system M_tot = M + m satisfies, by summing (5) and (6), the evolution equation

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 1/λ. To simplify the analysis, we will further assume that, initially, the system has already reached this state before the dynamics of aggregation starts. To do so, we chose the following unseeded initial conditions:

and thus, the total mass of the system is conserved for all time M_tot(t) = m₀. The term “unseeded” refers to the fact that, initially, there is no aggregated protein in the system (hence, no seed). This condition assumes a lack of aggregated species in a healthy in vivo state. Indeed, it is observed that soluble Aβ monomers are found in healthy individuals of all ages, while aggregates larger than monomers are generally correlated with Alzheimer’s disease progression.²² An extra advantage of this approach is that it fixes the constant γ = m₀λ.

Before we study the system in full generality, it is useful to consider the overall dynamics of the system for a typical set of parameters for the aggregation of Aβ40 given in the first column of Table I. We will use this set of parameters for all our examples. The other datasets are qualitatively equivalent, and the values of various derived quantities are given in Table I. As shown in Fig. 2, the typical behavior of the system from an unseeded initial condition is for the aggregate mass to increase up to a finite value M_∞, while the monomer concentration decreases to m_∞, in a typical sigmoid-like behavior. We observe that, in the absence of clearance, the monomer population is completely converted to aggregates (λ = 0, dashed curves in Fig. 2). Conversely, for large clearance, almost no conversion takes place (λ = 1, dotted curves in Fig. 2). Some of the monomers are converted (solid curves for λ = 0.2 in Fig. 2) for the case of moderate clearance. Of particular interest, for our discussion, is the change of behavior at some critical value λ_crit of the clearance λ where aggregation becomes negligible.

To derive a value for λ_crit, we determine the dependence of the asymptotic states m_∞ on λ. Considering steady state m = m_∞, P = P_∞, and M = M_∞ in (4) and (5), one expresses the latter two states as a function of parameters, λ and m_∞. These relations are substituted into (6) to produce the implicit equation H(m_∞, λ) = 0 with

For instance, for the same parameter values as in Fig. 2, we show in Fig. 3 the values of m_∞ as a function of λ. We observe a sharp transition for a critical value of the clearance parameter λ. There are three necessary conditions for λ_crit: first that λ_crit is non-negative, second that m_∞ is maximal, and third that the value of m_∞ coincides with m₀. The last two conditions can be realized by computing the derivative of the expression H(λ, m_∞) = 0 evaluated at m = m_∞. Therefore, λ_crit is given by the positive root of L(λ) = 0, where

For Aβ-40, the critical clearance, as shown in Fig. 3, is λ_crit = 0.72 h⁻¹. Critical clearance rates for the other experimental datasets are given in Table I for comparison.

In a neighborhood of λ_crit, m_∞, as a function of λ, undergoes a sharp transition. This transition is not a bifurcation in the strict sense, but, in the parlance of dynamical systems, it can be described as an imperfect transcritical bifurcation when heterogeneous nucleation and homogeneous nucleation terms can be understood as an imperfection and are sufficiently small with respect to the elongation. More specifically, when $k0/(k+m02)≪1$ and k_n/k₊ ≪ 1, the system is well approximated by k₀ = 0 and k_n = 0. In this limiting case, the fixed point (P, M, m) = (0, 0, m₀) for the system (4)–(6) undergoes a (perfect) transcritical bifurcation at $λ̃crit≈λcrit$ that can be obtained by locally expanding m_∞ in λ to find

where $λ̃crit$ is specified by the formula

and α is defined by the expression

When the clearance is close to the critical value, the linear approximation to the perfect bifurcation is a reasonable approximation for the imperfect bifurcation, as can be appreciated in Fig. 3 where $λ̃crit≈0.72$ and $m̃∞≈0.7+3.2λ̃$⁠. By analogy with epidemiology, we define a dimensionless neurodegenerative reproduction number,

such that for R₀ < 1 the aggregate level is negligible and grows to a finite value for R₀ > 1.

The existence of a critical clearance rate shows that in the healthy regime, i.e., for sufficiently large values of clearance, the system (1) and (2) with size-independent clearance can support a small, endemic, population of aggregates. The formation of a significant aggregate population, in this case, occurs only when the system’s clearance rate, λ, drops sufficiently below the critical clearance rate λ_crit. We can explore the dynamics close to the bifurcation by considering the normal form of the system for the perfect system [(4)–(6) with k₀ = k_n = 0] near $λ=λ̃crit$⁠. The general method to obtain the normal form of a transcritical bifurcation for an arbitrary smooth vector field is given in  Appendix A. Applying these ideas, we can approximate the full system by

where α is given by (13) and

Figure 4 shows a comparison of the total aggregate mass evolution, vs time, obtained for the imperfect unseeded system, the perfect seeded system, and the normal form. As expected, the agreement is excellent 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, which the data suggest is a good approximation to the system unless m_∞ approaches 0. Since we are interested in the asymptotic size distribution, we can assume that m = m_∞ in Eqs. (1) and (2), in which case, we have simply that

Using the definition of M = ∑_i>1ip_i, we obtain

This analysis is not valid for λ → 0. In that case, the total mass of the system is systematically transferred to larger and larger particles, and in the long-time limit, all finite aggregate concentrations tend to vanish and the trivial distribution is p_i = p_i−1 = p₂ = 0. However, in that limit, the assumption k_off = 0 is not justified anymore as even a small value of k_off allows for a non-trivial size distribution. Indeed, with k_off ≠ 0, we have the following recurrence relation for c_i:

with a single bounded solution of the form

with

An asymptotic expression of δ for small and large values of k_off gives

We see that unless λ = 0, the role of k_off, when sufficiently small, is negligible. We conclude that clearance (or depolymerization) is sufficient to obtain a non-degenerate size distribution (Fig. 5).

Next, we generalize the above treatment and allow the clearance rate of an aggregate to depend on its size. In this case, there is no simple, closed equation for the moments, as in Sec. III, and we must study the full system. Here, we make a key assumption about the dependence of the clearance on the aggregate size. We assume that there exists a critical aggregate size, N, such that all aggregates of size N, or greater, are too large to be cleared. Explicitly, this assumption implies that λ_i = 0, ∀i ≥ N. We also assume that k_off = 0 and n_c = n₂ = 2, and then (1) and (2) can be written as

where $P=∑i=2∞pi$ and $M̃=∑i=2∞ipi$⁠. The unseeded initial conditions for this system are

In general, under the assumption of a constant monomer production rate, there is no guarantee of mass conservation. For instance, if λ_i ≤ λ₁∀i > 2 and there is at least one i ≥ 2 such that λ_i < λ₁, then the overall mass of proteins will increase in time as shown in  Appendix B.

To study the dynamics of (25)–(28), we introduce a finite system with equivalent dynamics. Here, we follow Ref. 23 (see also Ref. 24) and introduce a super-particle, denoted q_N, which represents the concentration of all aggregates of size greater than or equal to N,

Since λ_i = 0 for all i ≥ N, we can take the limit of the partial sums of (28) to obtain

Since the monomer concentration m remains bounded, for any fixed time, the last term of (31) tends to zero as j → ∞ and the super particle concentration satisfies the equation

We will distinguish the finite system with a super-particle from the infinite system (25)–(28) by introducing the notation q_i = p_i for i < N. Defining $Q=∑i=2Nqi$⁠, and using (32), the corresponding super-particle system is defined by

The unseeded conditions for (33)–(37) are

For unseeded initial conditions, the dynamics of the finite system is equivalent to the infinite one in the following sense: First, note that $Q̇=Ṗ$⁠; this follows directly from the definition of Q, P, and (30). Thus, Q and P will agree for all time. In turn, (25) and (33) coincide when the initial data (29) and (38), respectively, are used; thus, $M̃(t)=M(t)$ in this case. Finally, by definition, p_i = q_i for 2 ≤ i < N and (30) and (31) have already established that solving (37) produces $qN(t)=∑i=N∞pi(t)$⁠, provided that the initial conditions agree. The above establishes an important fact that we rely on for the rest of the section; solving (25)–(28) with initial conditions (29) and (33)–(37) with initial conditions (38) yields

We remark, however, that M(t), defined as the solution of (33), is the total aggregate mass of both (25)–(28) and (33)–(37), due to (39), for the unseeded initial conditions (38); however, M(t) cannot be constructed a posteriori from the knowledge of q_i(t) where i = 2, 3, …, N in the same manner that $M̃(t)$ can be retrieved from the knowledge of p_i(t). That is, we have $M(t)≠∑i=2Niqi(t)$⁠. Indeed, in the closure process of reducing the full system to a finite one, we lost information regarding the mass of individual particles making up the superparticle. Nevertheless, both the evolution of aggregate mass of the full system and the size distribution (up to size N) can be obtained by studying the finite system (33)–(37).

Given a constant monomer production rate, systems such as (25)–(28) or (33)–(37), with size-dependent clearances, do not conserve mass in general (see  Appendix B) and the aggregate mass may increase with time. We study in more detail the particular choice

which expresses the modeling assumption that aggregates become increasingly difficult to clear as their size increases. An example of the dynamics of the system (33)–(37) is shown in Fig. 6. We observe two different behaviors. Initially, up to a time τ₂, the system mostly behaves like the conservative no-clearance model (λ = 0) even for large values of clearance. This behavior is markedly different than the one observed in Fig. 2. Second, for larger times, t > τ₂, the monomer mass always decreases and the aggregate mass always increases as predicted from our general analysis. We observe that larger clearance leads to faster aggregate mass creation. This is due to the fact that we balance monomer clearance by monomer production, so an increase in clearance automatically implies an increase in production. The question is then to understand the transition between the two regimes as well as the small and large time behaviors of all species. The situation where the clearance rate is higher relative to the production rate is shown in Fig. 7. Biologically, this corresponds, for example, to two individuals in which production proceeds at the same rate, but clearance is more efficient in one than in the other. While there is no qualitative change in behavior, i.e., the total concentration is still unbounded, both the speed at which aggregates accumulate and the steady state monomer concentration are lower in the system with a higher clearance rate. Thus, in a system with constant (i.e., time-invariant) monomer production rate and a clearance rate that decreases with size, whether increasing clearance will be beneficial or not will depend on whether it causes a simultaneous increase in the rate of monomer production or not. In either case, the aggregate mass is not bounded and only its rate of increase can be affected by changes in the clearance rate.

On long time scales, i.e., long enough so that the monomer concentration begins to decrease, the monomer production, aggregation, and nucleation processes result in an increase in subsequent aggregated species and, therefore, in the overall aggregate mass M. The asymptotic behavior of the system aggregate mass M is observed to depend entirely on the production rate, γ = λm₀, as

This behavior is illustrated in a log-plot in Fig. 8; the characteristic time scale, τ₂, indicates the time at which the monomer mass begins to decay. Once the asymptotic behavior of M has been established, the equations can be balanced asymptotically by the following dynamics:

where the symbol “∼” is understood as the long-time asymptotic behavior and 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 by the 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 clearance rate. Hence, we can obtain characteristic time scales for the amplification of the aggregate mass by considering the limit λ → 0⁺. In this case, the early evolution of the aggregate mass is governed by the dynamics of (4)–(6) with λ = 0. There are two characteristic time scales of importance. First, the time scale τ₁ associated with the exponential growth of the aggregate mass in early time via the inverse of the positive linear eigenvalue, μ = 1/τ₁, corresponding to the linearization of (4)–(6) around the healthy state m = m₀, M = P = 0. The linear eigenvalue is given by the positive root of

Second, there is a time scale τ₂ where both nucleation and amplification are balanced. It is given by the time for the linearized solution for M(t) to reach m₀. Hence, τ₂ is the solution of

For example, for the first parameter set (Aβ40) used for Figs. 6–8, these times are τ₁ ≈ 1.4 h and τ₂ ≈ 12.6 h. The value of τ₂ is a rudimentary estimate for the time of amplification; it is a lower bound for the typical time scale of growth (see Fig. 6). Nevertheless, in Fig. 8, we see that τ₂ can indeed act as an indicator for the onset of decay for the monomer mass. A more refined estimate can be obtained by using the approximate solution for the full dynamics given in Ref. 14.

In Secs. III and IV, we investigated a system in which the monomer production rate was constant, i.e., invariant with time. The other biologically reasonable assumption is that the monomer concentration instead remains at a constant level m₀. In this case, the system adjusts the monomer production rate to compensate for any loss of monomer, through degradation but also through aggregation. Experimental observations for example of the concentration of PrP, the main protein that aggregates in prion disease, in mice with prion disease support such a constant monomer.²⁵ We assume that, regardless of other parameters, (1) is instead specified by

so that, with unseeded initial conditions, we have m(t) = m₀ for all time. Assuming again no depolymerization, no fragmentation, and dimer nucleation, the master equations now read

where σ₀ = σ(m₀) and $M=∑i=2∞ipi$ is the total aggregate mass. This is an infinite system of linear ordinary 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 relations can be summarily presented by a power-law of the form

When ν = 0, we recover the size-independent case; when ν = −1, we recover the size-dependent diminishing clearance formulation used in Sec. IV; and finally, the case of ν = 1 corresponds to improved 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 state or increase indefinitely. The question is then to identify the critical values at which this transition happens.

We start with the simple case of size-independent clearance ν = 0; this is the analog to Sec. III for a constant free monomer assumption [cf. (45)] The moments (cf. Sec. III) are specified by a simple pair of linear equations given by

which can be written as

where q = (P, M)^T, $b=(k0+knm02,2k0+2knm02)T$⁠, and

The constant solution sole steady state for this system is q_∞ = −A⁻¹b; q_∞ is positive and finite if

This condition naturally provides a value for the critical clearance. Specifically, the largest linear eigenvalue for the system is $κ=λcrit(0)−λ$⁠; solutions converge to q_∞ exponentially in time (as e^κt) for $λ>λcrit(0)$ and grow unbounded for $λ≤λcrit(0)$ (see Fig. 9).

The values given in Table I for the different parameters show that this estimate is indistinguishable from the case studied in Sec. III, which is explained by 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 no longer size-independent. Then, the master equations do not yield a closed system for the moments. Nevertheless, due to the simplicity introduced by m(t) = m₀ being constant, we can find conditions for the existence of a fixed-point solution, $(p2*,p3*,…)$ to (46) and (47). If such a steady state $pi*$ for i > 2 exists, it must satisfy the recurrence relation

We note that each of the recursion coefficients, δ_i, is now dependent on i via λ_i. Define a sequence of real numbers, indexed by i, as

We define Δ₂ = 1, and the ith steady state is expressible, for all i ≥ 2, through its recurrence relation as

Defining

the steady state for the total aggregate mass solution M^* is then given by

and an application of (46), at steady state, gives the value of $p2*$ as

Therefore, for a fixed point to exist, we need the three following conditions to be satisfied:

An analysis of the case ν = 0 recovers the previous condition, and it can then be verified directly that conditions C1–C3 are satisfied, as expected, for $λ>λcrit(0)$⁠.

For ν = 1, we have (see  Appendix C) Δ⁽¹⁾ = 2 + b/λ, and the steady population of dimers, whenever it exists, is given by

Hence, condition C3 leads to $λ>λcrit(1)$ with

We note that the above implies that the critical clearance depends only on the secondary nucleation process and, in particular, not the process of elongation [cf. $λcrit(0)$ in (53)].

For ν = −1, the situation is not as simple. Condition C1 is verified, but C2 leads to λ > 2b for which

where Γ(·) is the usual Gamma function. Condition C3 is satisfied if $λ>λcrit(−1)$ where $λcrit(−1)$ is the positive solution of