Table 2 Time evolution of the fibril number concentration (P), fibril mass concentration (M) and oligomer mass concentration (S) for the microscopic processes involved in amyloid formation [181]. m(t), time-dependent monomer concentration; M(t), time-dependent fibril mass concentration; P(t), time-dependent fibril number concentration; S(t), time-dependent oligomer mass concentration; kn, primary nucleation rate constant; k2, secondary nucleation rate constant; k+, elongation rate constant; k-, fragmentation rate constant; kd, oligomer dissociation rate constant; ko1, primary oligomer association rate constant; ko2, secondary oligomer association rate constant; kconv, oligomer conversion rate constant; kd2, fibril-mediated oligomer dissociation rate constant; nc, reaction order of primary nucleation; n2, reaction order of secondary nucleation; nconv, reaction order of oligomer conversion; no1, reaction order of primary oligomer association; no2, reaction order of secondary oligomer association; KE, Michaelis–Menten constant for saturating elongation (monomer concentration at which the rate of elongation is half the maximal velocity, Vmax); KM (Michaelis–Menten constant for saturating secondary nucleation (monomer concentration at which the rate of secondary nucleation is half the maximal velocity, Vmax)
Molecular process | Time evolution model | Illustration of the model |
|---|---|---|
Primary nucleation | \(\frac{dP}{dt}={k}_{n}m{\left(t\right)}^{{n}_{c}}\) |
|
Heterogeneous primary nucleation | varies |
|
Elongation | \(\frac{dM}{dt}=2{k}_{+}m\left(t\right)P\left(t\right)\) |
|
Saturating elongation | \(\frac{dM}{dt}=\frac{2{k}_{+}m\left(t\right)P\left(t\right)}{1+\frac{m\left(t\right)}{{K}_{E}}}\) |
|
Fragmentation | \(\frac{dP}{dt}={k}_{-}M\left(t\right)\) |
|
Secondary nucleation | \(\frac{dP}{dt}={k}_{2}m{\left(t\right)}^{{n}_{2}}M\left(t\right)\) |
|
Multistep secondary nucleation | \(\frac{dP}{dt}=\frac{{k}_{2}m{\left(t\right)}^{{n}_{2}}M\left(t\right)}{{\left(1+\frac{m\left(t\right)}{{K}_{M}}\right)}^{{n}_{2}}}\) |
|
Primary association of oligomers | \(\frac{dS}{dt}={k}_{o1}m{\left(t\right)}^{{n}_{o1}}\) |
|
Primary dissociation of oligomers | \(\frac{dS}{dt}={k}_{d}S\left(t\right)\) |
|
Oligomer conversion | \(\frac{dS}{dt}={k}_{conv}m{\left(t\right)}^{{n}_{conv}}S\left(t\right)\) |
|
Secondary association of oligomers | \(\frac{dS}{dt}={k}_{o2}m{\left(t\right)}^{{n}_{o2}}M\left(t\right)\) |
|
Fibril-mediated oligomer dissociation | \(\frac{dS}{dt}={k}_{d2}S\left(t\right)M\left(t\right)\) |
|
