## 3 Results
### Comparison of A\(\beta\) Accumulation with Its Fibr Formation  
A\(\beta\) fibril growth consists of nucleation and subsequent elongation [28]. Further, induction of fibril growth depends on the environment [24; 29]. Nucleation of A\(\beta\) and its elongation requires its accumulation. The number density of A\(\beta\) fibrils formed on each membrane was compared with the corresponding \(\Delta f\) value (Figure 5A) to confirm the effect of planar membranes on A\(\beta\) fibril growth. Increment in \(\Delta f\) increased the number density of the fibrils. It was therefore considered that the accumulation of A\(\beta\) resulted in its nucleation. It must be noted that the process involves primary and secondary nucleation [28]. Primary nucleation refers to the nucleation process that occurs in a previously fibril-free solution. In contrast, secondary nucleation is the generation of new fibrils by fibrils already present in the suspension, which is associated with the variation in the number density of fibrils [28]. The present number density of fibrils was attained following both nucleation steps. In this study, the DOPC- and PSf-based membranes indicated a definite variation in the number density of fibrils after the increase in ThT fluorescence intensity (Figure 3A,E). Therefore, primary nucleation is addressed in this study.  
Figure 5: **(A)** Relationship between the accumulated A\(\beta\) (\(\Delta f\)) and its nucleation rate. **(B)** Relationship between the \(\Delta f\) value and the number density of A\(\beta\) fibrils. PLLA-5k- and PLLA-10k-based membranes were ruled out in **(A,B)**. **(C)** Relationship between the molecular weight of polymers and the nucleation rate. Closed and open keys represent phospholipid- and polymer-based membranes, respectively.  
The rate of primary nucleation can be defined as the reciprocal of the lag time. Accordingly, the nucleation rate of A\(\beta\) was compared with the \(\Delta f\) value (Figure 5B). The \(\Delta f\) value was linearly proportional to the nucleation rate except for the DPPC-based membrane, i.e., the accumulation of A\(\beta\) on planar membranes accelerated its primary nucleation. Thus, the interaction between A\(\beta\) and the planar membranes determines the rate of A\(\beta\) nucleation.  
The interaction between A\(\beta\) and planar membranes is affected by the hydrophobicity of the polymers as well as by the entanglement of polymers [34]. As shown in Table 1, the entanglement depends on the \(Mw\) of the polymers (see the possible entanglement \(Mw/M_{\text{e}}\)). The \(Mw\) of a polymer also regulates fibril growth [29]. [PERSON] stated that condensed-fluid membranes of lipids behave similarly to linear-flexible polymers [51]. Thus, the rates of A\(\beta\) nucleation were compared with the \(Mw\) of polymers, by adding the polymers, except PLLA, together with phospholipids (Figure 5C). The increase in \(Mw\) of the polymer reduced the nucleation rate, except for PEG-PE. It is likely that the deviation of the PEG-PE-based membrane originated from the entanglement by PEG chains. Notably, a correction considering the disruption or detachment of PLLA-5k- and PLLA-10k-based membranes due to their interaction with A\(\beta\) (Figure 4B) would be needed. Thus, an entanglement between polymers and/or between A\(\beta\) and the polymer is likely to directly impact the primary nucleation process.