arXiv subjects

Alexander J. Dear

Publications and source records attributed to Alexander J. Dear.

4 recordsLinked to original sources

arXiv2025

Kinetics of seeded protein aggregation: theory and application

``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.

physics.chem-ph↗

arXiv2023

General integrated rate law for complex self-assembly reactions reveals the mechanism of amyloid-beta co-aggregation

Analyzing kinetic experiments on protein aggregation using integrated rate laws has led to numerous advances in our understanding of the fundamental chemical mechanisms behind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases. However, the description of biologically relevant processes may require rate equations that are too complex to solve using existing methods, hindering mechanistic insights into these processes. An example of significance is co-aggregation in environments containing multiple amyloid-beta (Abeta) peptide alloforms, which may play a crucial role in the biochemistry of Alzheimer's disease but whose mechanism is still poorly understood. Here, we use the mathematics of symmetry to derive a general integrated rate law valid for most plausible linear self-assembly reactions. We use it in conjunction with experimental data to determine the mechanism of co-aggregation of the most physiologically abundant Abeta alloforms: Abeta42, Abeta40, Abeta38 and Abeta37 peptides. We find that Abeta42 fibril surfaces catalyze the formation of co-oligomers, which accelerate new Abeta40, Abeta38 and Abeta37 fibril formation whilst inhibiting secondary nucleation of new Abeta42 fibrils. The simplicity, accuracy and broad applicability of our general integrated rate law will enable kinetic analysis of more complex filamentous self-assembly reactions, both with and without co-aggregation.

physics.chem-ph↗

arXiv2023

Approximate Lie symmetries and singular perturbation theory

Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact that all exact solutions of differential equations are consequences of (Lie) symmetries, we reformulate perturbation theory for differential equations in terms of expansions of the Lie symmetries of the solutions. This is a change in perspective from the usual method of obtaining series expansions of the solutions themselves. We show that these approximate symmetries are straightforward to calculate and are never singular; their integration is therefore an easier way of constructing uniformly convergent solutions. This geometric viewpoint naturally subsumes the RG-inspired approach of Chen, Goldenfeld and Oono, the method of multiple scales, and the Poincare-Lindstedt method, by exploiting a fundamental class of symmetries that we term ``hidden scale symmetries''. It also clarifies when and why these singular perturbation methods succeed and just as importantly, when they fail. More broadly, direct, algorithmic identification and integration of these hidden scale symmetries permits solution of problems where other methods are impractical.

math-ph↗

arXiv2019

Geometric localization in supported elastic struts

Localized deformation patterns are a common motif in morphogenesis and are increasingly finding widespread applications in materials science, for instance as memory devices. Here we describe the emergence of spatially localized deformations in a minimal mechanical system by exploring the impact of growth and shear on the conformation of a semi-flexible filament connected to a pliable shearable substrate. We combine numerical simulations of a discrete rod model with theoretical analysis of the differential equations recovered in the continuum limit to quantify (in the form of scaling laws) how geometry, mechanics, and growth act together to give rise to such localized structures in this system. We find that spatially localized deformations along the filament emerge for intermediate shear modulus and increasing growth. Finally, we use experiments on a 3D printed multi-material model system to demonstrate that external control of the amount of shear and growth regulates the spatial extent of the localized strain texture.

cond-mat.soft↗