. 2019 Sep 25;8:e46574. doi: 10.7554/eLife.46574

Table 4. Reaction schemes, rate equations and rate constants for the fibril elongation models tested.

X represents the species that add onto the fibril ends.

Module	Variant	Reaction scheme	Rate equations	Rate constants
Pre-polymerization	No Pre-polymerization (Monomer addition)	$X = X_{1}$	$\frac{d [X]}{d t} = \sum_{i = 2}^{N} {- k}_{e} F_{i - 1} X_{} + k_{e}^{'} F_{i}$	$k_{1} = k_{e}$ $k_{1}^{'} = k_{e}^{'},$
	Monomer conformational exchange	$X_{1} \overset{k_{1}}{\underset{k_{1}}{⇄}} X_{1}^{'}$ $X = X_{1}^{'}$	$\frac{d [X_{1}]}{d t} = - k_{1} X_{1} + k_{1}^{'} X_{1}^{'}$ $\frac{d [X]}{d t} = {\begin{matrix} k_{1} X_{1} - k_{1}^{'} X_{1}^{'} \\ + \sum_{i = 2}^{N} - k_{e} F_{i - 1} X + k_{e}^{'} F_{i} \end{matrix}$	$k_{1}, k_{e}$ $k_{1}^{'}, k_{e}^{'},$
	Dimer addition	$X_{1} + X_{1} \overset{k_{1}}{\underset{k_{1}}{⇄}} X_{2}$ $X = X_{2}$	$\frac{d [X_{1}]}{d t} = - 2 k_{1} X_{1} X_{1} + 2 k_{1}^{'} X_{2}$ $\frac{d [X]}{d t} = {\begin{matrix} k_{1} X_{1} X_{1} - 2 k_{1}^{'} X \\ + \sum_{i = 2}^{N} - k_{e} F_{i - 1} X + k_{e}^{'} F_{i} \end{matrix}$	$k_{1}, k_{e}$ $k_{1}^{'}, k_{e}^{'},$
	Hexamer addition	$X_{1} + X_{1} \overset{k_{1}}{\underset{k_{1}^{'}}{⇄}} X_{2}$ $X_{2} + X_{2} + X_{2} \overset{k_{1}}{\underset{k_{1}^{'}}{⇄}} X_{6}$ $X = X_{6}$	$\frac{d [X_{1}]}{d t} = - 2 k_{1} X_{1} X_{1} + 2 k_{1}^{'} X_{2}$ $\begin{array}{ll} \frac{d [X_{2}]}{d t} = k_{1} X_{1} X_{1} - k_{1}^{'} X_{2} - \\ 3 k_{2} X_{2} X_{2} X_{2} + 3 k_{2}^{'} X_{6} \end{array}$ $\frac{d [X]}{d t} = {\begin{matrix} k_{2} X_{2} X_{2} X_{2} - k_{2}^{'} X \\ + \sum_{i = 2}^{N} - k_{e} F_{i - 1} X + k_{e}^{'} F_{i} \end{matrix}$	$k_{1}, k_{2}, k_{e}$ , $k_{1}^{'}, k_{2}^{'}, k_{e}^{'},$
	Monomer-Dimer-Tetramer-Hexamer	$X_{1} + X_{1} \overset{k_{1}}{\underset{k_{1}^{'}}{⇄}} X_{2}$ $X_{2} + X_{2} \overset{k_{2}}{\underset{k_{2}^{'}}{⇄}} X_{4}$ $X_{2} + X_{4} \overset{k_{3}}{\underset{k_{3}^{'}}{⇄}} X_{6}$	$\frac{d [X_{1}]}{d t} = - 2 k_{1} X_{1} X_{1} + 2 k_{1}^{'} X_{2}$ $\begin{array}{ll} \frac{d [X_{2}]}{d t} = k_{1} X_{1} X_{1} - k_{1}^{'} X_{2} - \\ 2 k_{2} X_{2} X_{2} + 2 k_{2}^{'} X_{4} - \\ k_{3} X_{4} X_{2} + k_{3}^{'} X_{6} \end{array}$ $\begin{array}{ll} \frac{d [X_{4}]}{d t} = k_{2} X_{2} X_{2} - k_{2}^{'} X_{4} - \\ k_{3} X_{4} X_{2} + k_{3}^{'} X_{6} \end{array}$	$k_{1}, k_{2}, k_{3} k_{e}$ $k_{1}^{'}, k_{2}^{'}, {k_{3}^{'}, k}_{e}^{'},$
		$X = X_{6}$	$\frac{d [X]}{d t} = {\begin{matrix} k_{3} X_{4} X_{2} - k_{3}^{'} X \\ + \sum_{i = 2}^{N} - k_{e} F_{i - 1} X + k_{e}^{'} F_{i} \end{matrix}$
Polymerization		$\begin{array}{ll} X X \\ ↓ ↓ \\ F \overset{k_{e}}{\underset{k_{e}^{'}}{⇄}} F_{1} \overset{k_{e}}{\underset{k_{e}^{'}}{⇄}} F_{2...} F_{N} \\ ↓ ↓ \\ X X \end{array}$	$\frac{d [F]}{d t} = - k_{e} X_{} F_{1} + k_{e}^{'} F_{2}$ $\begin{array}{ll} \frac{d [F_{i}]}{d t} = k_{e} X_{} F_{i - 1} - k_{e}^{'} F_{i} - k_{e} \\ X_{} F_{i} + k_{e}^{'} F_{i + 1} 2 \leq i < N \end{array}$ $\frac{d [F_{N}]}{d t} = k_{e} X_{} F_{i - 1} - k_{e}^{'} F_{i}$