## 3 Results
### Kinetic Analysis of \(A\beta\) Fibril Growth on Various Membranes  
Variations in the ThT-stained aggregates were monitored for various membranes. To discuss the effect of planar membranes on the kinetic properties of fibril growth, the ThT intensity was plotted against the incubation time for four typical membranes (Figure 3A). The planar membranes showed sigmoidal curves. ThT fluorescence intensity increased at approximately 10 h in the PSF-based membrane, whereas the PVP-based membrane showed an increase in ThT fluorescence intensity after 80 h. The lag time in the sigmoidal curve corresponds to the nucleation period, and a subsequent increase in ThT intensity represents the elongation of the nuclei [28]. The kinetic parameters associated with the lag time were subsequently assessed using Equation (2):  
\[I(t)=I_{\mathrm{i}}+m_{\mathrm{i}}t+(I_{\mathrm{max}}+m_{\mathrm{f}}t)/(1+ \exp\{-\left(t-t_{\mathrm{m}}\right)/\uptau\}), \tag{2}\]  
where \(t_{\mathrm{lag}}\) (h) is the lag time, \(k_{\mathrm{app}}\) (=1/\(\uptau\)) (h\({}^{-1}\)) is the apparent elongation rate constant, \(I_{\mathrm{i}}\) and \(I_{\mathrm{max}}\) are the ThT intensities at the initial and final stages, respectively, and \(m_{\mathrm{i}}\) and \(m_{\mathrm{f}}\) are the correction factors at the initial and final stages, respectively. The \(t_{\mathrm{m}}\) value is associated with the \(t_{\mathrm{lag}}\) value as follows: \(t_{\mathrm{lag}}=t_{\mathrm{m}}-2k_{\mathrm{app}}-^{1}\). Thus, the sigmoidal curve shown in Figure 3A can be characterized by \(k_{\mathrm{app}}\), \(t_{\mathrm{lag}}\), and \(I_{\mathrm{max}}\). The \(k_{\mathrm{app}}\) values were plotted against the corresponding \(t_{\mathrm{lag}}\) values (Figure 3B). Overall, a negative correlation was observed between both. This suggests that the fibril formed after the fast nucleation favored fast elongation. In other words, both processes are linked to each other. Figure 2: TIRFM images of ThT-stained amyloid fibrils of A\(\beta\) on various planar membranes: (**A**) control; (**B**) DOPC; (**C**) iso-PMMA; (**D**) syn-PMMA; (**E**) PVP; (**F**) PSf; (**G**) PLLA-50k. (**G**’) TEM image of Figure 3G. (**H**) SEM image of PLLA-50k-based membrane. The \(t_{\rm lag}\) values observed for various planar membranes are shown in Figure 3C. From a comparison of PSf- (-10 h) with PVP-based membranes (-120 h), the \(t_{\rm lag}\) value was likely strongly dependent on the hydrophobicity of the polymers. The same was true for PEG-PE-based membranes (-100 h) considering the induction of hydrophilic PEG chains into phospholipids, such as DOPC and DPPC. The \(t_{\rm lag}\) values for _iso_-PMMA- and _syn_-PMMA-based membranes were relatively greater than those for the other membranes, although the influence of tacticity in PMMA was not significant, suggesting that these two membranes were disadvantageous for the nucleation of A\(\beta\), as compared with other membranes. In addition, there was no definite influence of the entanglement of polymers on the lag time for PLLAs with different _Mw_. The \(I_{\rm max}\) values, one of the quantitative indices for fibrillogenesis [28], are shown in Figure 3D. \(I_{\rm max}\) for PEG-PE- and PVP-based membranes was very low. PSf-based membranes exhibited high \(I_{\rm max}\) values relative to those of PVP-based membranes. The same was true for DOPC- and DPPC-based membranes relative to the PEG-PE-based membrane. Therefore, the \(I_{\rm max}\) value clearly depends on the hydrophobicity of the planar membranes. The tacticity of PMMA was not significantly observed in \(I_{\rm max}\). Alternatively, the \(I_{\rm max}\) value for PLLA increased with an increase in molecular weight, suggesting a possible influence of the entanglement of polymers. According to a previous report [28], \(I_{\rm max}\) is determined by the number of fibrils formed and/or their nanoscopic structure to which ThT binds. The number density of fibrils after the growth of fibrils was counted subsequently.