A simple method for modeling amyloid kinetics featuring position biased fiber breakage

Abstract

Significance: 'A simple method for modeling amyloid kinetics featuring position biased fiber breakage' A kinetic model of amyloid formation is presented based on two inter-related ordinary differential equations. The model can simply account for different rates of breakage at the fiber ends vs. internal regions. In the former case, breakage releases a monomer unable to maintain its amyloid structure. In the latter case, breakage produces two smaller fibers each able to act as 'seeds' able to facilitate amyloid growth. Thus position dependent susceptibility to breakage could determine whether breakage encourages or discourages amyloid growth. These outcomes are crucial when considering the role amyloid growth plays in disease aetiology, such as for Alzheimer's disease.Abstract: A mathematical model of amyloid fiber formation is described that is able to simply specify different rates of fiber breakage at internal versus end regions. This Note presents the derivation of the relevant equations and provides results showing the dramatic effects of position biased fiber breakage on the kinetics of amyloid growth.Protein helical polymerization reactions, such as those constituted by actin and tubulin assembly, have been a much studied topic in biophysics [1,2]. Amyloid, a fibril-like homopolymer capable of being formed from many different protein monomer building blocks, has been shown to exhibit both helical polymerization kinetics and steady state behavior capable of quantitative description using helical polymerization models of Oosawa and colleagues [3][4][5][6]. A number of important extensions to these models been developed over the last twenty years to better describe amyloid formation [7][8][9][10][11][12][13][14]. These extensions are based on a consensus chemical schema (Fig. 1) featuring the following three general processes, (i.) Reversible fiber nucleation; (ii.) Fiber growth and dissolution via monomer addition and loss; and (iii) Fiber breakage and joining. A simple kinetic description containing some aspects of the consensus mechanism was developed by Smith et al. [15] who employed an approach cast in terms of average quantities. Although providing less information about the time dependent evolution of the polymer distribution, this reduced model proved particularly simple to code and produced easily interpretable output. However, due to the approximations involved in its derivation, the Smith et al. [15]

MIL Cell – A tool for multi-scale simulation of yeast replication and prion transmission

“…The kinetic models implemented in MIL-CELL are cast in terms of experimentally observed mechanisms previously demonstrated to have relevance to biology (e.g. variable internal versus endwise amyloid breakage relationships [Hall, 2020] , various nucleation and growth relationships [Nowak et al 1999; Pallitto and Murphy, 2001; Hall and Edskes, 2004, 2009, 2012; Hall and Hirota, 2009; Hirota et al 2019] and various higher order (end-to-end or lateral joining) amyloid association [Zhao et al 2016] . Another important and distinctive feature of the current work is the specifiable component partition rate between mother and daughter cells during cell division (Eqn.…”

“…To simulate preferential endwise depolymerization of amyloid filaments (the so called endwise dissolution or ‘munching’ case postulated by Zhao et al 2018) a previously developed mathematical model, able to account for position dependent differences in fiber fracture rate i.e. b G ≠ b A1 , was utilized [ Hall, 2020 ]. Limited to the consideration of single filament growth this model is capable of describing the time dependence of monomer formation ( Eqn.…”

“…9e ) for which the decay constant is calculated on the basis of knowledge of the average polymer degree ( Eqn. 9d,f ) [Hall 2020] . With the basic form of the kinetic equations for amyloid growth occurring in a fixed volume we now describe the functionalization of the partition term.…”

“… (B) Differential ‘Munching’ of amyloid fibers - in which the intrinsic rates of amyloid breakage are considered to occur differently at the end of the polymer and at internal sites [e.g. Hirota et al 2019; Hall, 2020] {in this particular example internal breakage is considered greater than endwise depolymerization b N = b G = 0s −1 ; b A1 = 0.005s −1 ; f G = f A1 rate constant b A1 (units s −1 )). Growth and shrinkage of a clumped amyloid fiber of arbitrary degree x+y {indicated by a cross bridge existing between the grey squares of two amyloid single fibers of individual degree x and y) is set by second order forward, f A2 (units M −1 s −1 ) and backward, b A2 (units s −1 ) rate constants.…”

A fortunate period of overlap with Prof. Haruki Nakamura

“…My research work there involved investigating amyloid formation in yeast with the goal of both modelling this process computationally and solving the three-dimensional structure of the amyloid prion via cryo-electron microscopy. My work situation at the NIH was scientifically ideal (resulting in a few high-quality publications (Hall 2020a , b ) and another edited special Issue (Hall et al 2020 ). My position also allowed me sufficient time and space to engage with my newfound appointment as Chief Editor of the Biophysical Reviews journal which occurred in 2019.…”