Kinetics of seeded protein aggregation: theory and application

TL;DR

This work introduces a generalized, analytically tractable framework for seeded protein aggregation that remains accurate at moderate to high seed levels by employing asymptotic Lie symmetry methods. It yields a universal analytical solution for seeded, saturable kinetics and provides concrete expressions for the seed levels needed to bypass primary or both primary and secondary nucleation, as well as how saturation shapes average fibril lengths. The authors extend the theory to practical experimental design, including how seed length and fragmentation influence elongation-dominated regimes, and offer guidance for accurate kinetic fitting and interpretation. Together with updated tools for data fitting, the approach enables more precise disentanglement of nucleation and elongation contributions in amyloid kinetics and informs seed preparation strategies and fragmentation methods for experimental studies.

Abstract

``Seeding'' is the addition of preformed fibrils to a solution of monomeric protein to accelerate its aggregation into new fibrils. It is a versatile and widely-used tool for scientists studying protein aggregation kinetics, as it enables the isolation and separate study of discrete reaction steps contributing to protein aggregation, specifically elongation and secondary nucleation. However, the seeding levels required to achieve dominating effects on each of these steps separately have been established largely by trial-and-error, due in part to the lack of availability of integrated rate laws valid for moderate to high seeding levels and generally applicable to all common underlying reaction mechanisms. Here, we improve on a recently developed mathematical method based on Lie symmetries for solving differential equations, and with it derive such an integrated rate law. We subsequently develop simple expressions for the amounts of seed required to isolate each step. We rationalize the empirical observation that fibril seeds must often be broken up into small pieces to successfully isolate elongation. We also derive expressions for average fibril lengths at different times in the aggregation reaction, and explore different methods to break up fibrils. This paper will provide an invaluable reference for future experimental and theoretical studies in which seeding techniques are employed, and should enable more sophisticated analyses than have been performed to date.

Figures (8)

• Figure 1: The key reaction steps in protein aggregation mechanisms and how they are affected by seed. a: Primary nucleation is the formation of fibrils from monomers in the absence of seed. b: With low initial seed concentrations, primary nucleation is not a significant source of new fibrils at any time. However, secondary nucleation remains responsible for the majority of new fibril formation over the entire reaction. c: At high seed levels, only elongation of existing fibrils is important for the overall kinetics.
• Figure 2: In the absence of elongation saturation, the kinetics of protein aggregation can be partitioned into two asymptotic regimes: $\mu\to 1$ and $\mu\to 0$. Parameters: $n_2=3,\ n_c=2,\ \varepsilon=0.01$. a-b: numerical trajectory for $\Pi$ can be successfully normalized to a maximum of 1 by dividing by $\Pi_\infty$ as calculated in Eq. \ref{['Piinfty']}, validating our analytical solution for $\Pi_\infty$. a,c: Unsaturated kinetic curves ($K_S=10m_\text{tot}$) for $\Pi$ (numerical solution) and $M/m_\text{tot}$ (black: numerical solution). The point at which $\Pi=0.8\Pi_\infty$ is a good indicator of when we transition from the $\mu\to 1$ asymptotic regime for the kinetics to the $\mu\to 0$ regime. This is demonstrated by the $\mu\to 1$ asymptotic solution (Eq. \ref{['genseries1']}, red) losing accuracy and the $\mu\to 0$ asymptotic solution (Eq. \ref{['mu3soln']}, cyan, dashed) becoming an accurate approximation to the overall kinetics. b,d: Saturated ($K_S=0.04m_\text{tot}$) kinetic curves for $\Pi$ and $M/m_\text{tot}$. As expected, the transition between asymptotic regimes comes later, at lower $\mu$, due to the lower $\mu_c$ at which $\Pi$ stops growing significantly.
• Figure 3: Comparison of Eq. \ref{['gensolnsat']} to earlier solutions for the kinetics of linear protein self-assembly. a: Eq. \ref{['gensolnsat']} offers a modest improvement in accuracy over earlier solutions when describing unseeded kinetics with no fibrils present at $t=0$. Parameters: $n_2=5$, $n_c=3$, $\varepsilon=0.04$ and no saturation. b: Earlier solutions fail to describe adequately the kinetics when moderate concentrations of fibrils are present at $t=0$. By contrast, Eq. \ref{['gensolnsat']} provides almost exact results. Parameters: $n_2=5$, $n_c=4$, $\varepsilon=0.1$, $\Pi(0)=2.16$, $\mu(0)=0.45$ and no saturation.
• Figure 4: The evolution of average fibril lengths (number of monomeric subunits) over time is well captured by our analytical solutions and has a complex dependence on the reaction mechanism. Initially new fibrils are formed mainly through primary nucleation, and the average length always increases with time. Once the fibril mass becomes high enough that secondary processes take over as the main fibril formation mechanism, the average fibril length achieves steady-state, given by Eq. \ref{['eq:lss']}. a: When the mechanism is unsaturated ($K_X=100m_\text{tot}$ for all steps) and features secondary nucleation, the fibrils start to increase in length from the steady-state value towards final value Eq. \ref{['Linf_nosat']} once significant depletion of monomer begins to occur. This is because fibril formation reduces more rapidly with monomer concentration than elongation. b: Conversely, saturation in secondary nucleation ($K_S/m_\text{tot}=0.1$) causes fibril lengths to reduce after the steady-state length has been attained, since now elongation is more sensitive to monomer depletion. For the same $v_\text{max}$, the final average fibril length (Eq. \ref{['Linf_secsat']}) is thus lower. c: Saturation in elongation ($K_E/m_\text{tot}=0.1$) causes a more pronounced growth in fibril length upon monomer depletion, reaching a higher final value (Eq. \ref{['Linf_elsat']}), as the relative sensitivity of secondary nucleation to falling monomer levels becomes more pronounced. Parameters: A40 reaction orders and rate constants Meisl2014, modified to ensure the same $v_\text{max}=k_xK_X^{n_x}$ in each panel as in Meisl2014. Longer times spent at $L_\text{ss}$ are in principle achieved in systems with smaller $\varepsilon$.
• Figure 5: Comparing the the aggregation kinetics with secondary nucleation (black) with a hypothetical reaction without secondary nucleation (purple, dashed), for a number of different seed lengths and seed concentrations. Kinetic curves are simulated by numerical integration of Eqs. \ref{['momeqs']}. Top row: 30% seed (0.3 $\mu$M), length of fibrils is that present at the end of a reaction starting from the same monomer concentration, without seed. Middle row: seed concentration is $M_{crit,2}$, length of fibrils is that present at the end of a reaction starting from the same monomer concentration, without seed. Bottom row: 30% seed (0.3 $\mu$M), length of fibrils is $L_{crit,2}$. The left column assumes no saturation, the middle column saturation of elongation only, and the right column saturation of secondary nucleation only. Parameters used: $m_0 = 1$, $k_+ = 10^4$, $k_2 = 3.6\cdot10^{-5}$ (black) or $k_2=0$ (purple, dashed), $n_2 = 2$, $K_E = 0.1$ (relevant only in middle column) and $K_S = 0.1$ (relevant only in right column), with units in $\mu$M and h in all cases.