4.3.2. Correlation Relation between ΔpHo and dpH and Extraction of Coverage Ratio

As the protein covers more of the colloidal surface, a less negative net ionic charge can be achieved. In other words, the negative charge is partially quenched due to the coverage of the peptide over the gold surface. This indicates that a less acidic condition is required to neutralize the colloidal surface.

Assuming a linear relationship (i.e., y = mx = b, where m is a slope and b is an intercept), all data points are fit along the formula given by *d*pH = *m* ΔpHo + *b*, where *d*pH corresponds to y and ΔpHo corresponds to x in the above linear relationship as shown in Figure 13. We hypothesize that ΔpHo directly relates to the peptide coverage fraction, Θ, since ΔpHo exhibits the surface character change between peptide coated gold colloid and bare gold colloid. Thus, ΔpHo = 0 corresponds to Θ = 0 (i.e., no peptide coverage or bare gold colloid), and the x-axis intercept of *d*pH = *m* ΔpHo + *b* indicates the maximum value of <sup>Δ</sup>pHo, which occurs when <sup>Θ</sup> <sup>=</sup> 1 at *<sup>d</sup>*pH <sup>=</sup> 0 (i.e., <sup>λ</sup>peak (1) <sup>~</sup>∞). The maximum coverage, therefore, can be achieved at the <sup>Δ</sup>pHo value given by <sup>Δ</sup>*pHo*(*max*) = <sup>−</sup> *<sup>b</sup> <sup>m</sup>* . By replacing ΔpHo (max.) with Θ = 1, any Θ values in between ΔpHo = 0 and ΔpHo (max.) can then be calculated by

$$
\Theta = \frac{\Delta pH\_o}{\Delta pH\_o(\max)}\tag{2}
$$

**Figure 13.** For each amyloidogenic peptide, *d*pH obtained by Equation (1) by analyzing each sigmoidal plot (See Figure 12b) was plotted as a function of ΔpHo. The coverage ratio, Θ, for each gold colloidal size was obtained by scaling the data point, based on the x-axis intercept Θ =1 (See Equation (2)) Each data point corresponds to different nano-gold colloidal size and was obtained by analyzing sigmoidal plot shown in Figure 12b) for bare gold and amyloidogenic peptide coated gold.

#### 4.3.3. Simulation Process for Calculating the Coverage Fraction

A full explanation of a simulation procedure of calculating peptide coverage fraction, Θ, will be described in a report by Yokoyama and Ichiki [54]. In this paper, an essential concepts for calculating Θ are briefly presented.

The adsorption orientation of a prolate (axial length of *a* and *b*, *a* < *b*) was selected from either spiking-out orientation (Figure 14a top) or lie-down orientation (Figure 14a bottom). In spiking-out orientation, *b*-axis of a prolate contacts tangentially at a nano-gold surface. On the other hand, in lie-down orientation, *a*-axis of a prolate contacts tangentially at a nano-gold surface. An area projected on the sphere surface as an occupying area (Asphere) is Asphere = π*a*<sup>2</sup> for spiking-out orientation and Asphere = π*ab* for lie-down orientation. Once adsorption orientation was selected, the numbers of adsorption points were calculated for the first layer and the second layer.

For the first layer, the total adsorption points (*nf,*tot) is a sum of a top and a bottom spot of a sphere (*nf,*top and *nf,*bot), equatorial spots (*nf,*eq), and both hemi-sphere's axial position for each equatorial spot j (*nf,*ax,j) as:

$$n\_{f, \text{tot}} = n\_{f, \text{top}} + n\_{f, \text{bot}} + n\_{f, \text{eq}} + 2 \sum n\_{f, \text{ax}, j} \tag{3}$$

As for the second layer, avoiding the adsorption points taken by the first layer, the total adsorption points (*ns,*tot) are given by

$$n\_{\rm s, tot} = n\_{\rm s,eq} + 2\sum n\_{\rm s,ax,j} \tag{4}$$

where equatorial spots (*ns,*eq), and both hemi-sphere's axial position for each equatorial spot j (*ns,*ax,j) are counted over a sphere corresponds to a second layer. The axial length *a* and *b* (*a* > *b*) of an approximated prolate for Aβ1–40 [17], α-synuclein [37] and β2m [38] are estimated to be: Aβ1–40 (*a*, *b*) = (2.1 nm, 4.1 nm), α-syn (*a*, *b*) = (2.9 nm, 6.0 nm), and β2m (*a*, *b*) = (2.1 nm, 4.6 nm). Each colloidal particle is approximated to be a sphere with diameter *d* (radius *r* = *d*/*2*).

In order to reproduce observed Θ, Θobs (= Θtotal), a combination of the first and second layers were calculated by Equation (5).

$$
\Theta\_{\text{total}} = \Theta\_{\text{f, total}} + \gamma \,\Theta\_{\text{s, total}} \tag{5}
$$

where Θ*f*, total is a fraction of coverage contributed from the first layer, Θ*s*, total is a fraction of coverage contributed from the second layer, and γ is an empirical factor indicating the weight of the second layer. Each of Θ*f*, total and Θ*s*, total is determined by Equation (6) and Equation (7), respectively.

$$\Theta\_{f, \text{total}} = \frac{\mathbf{A}\_{\text{product}} \times \mathbf{n}\_{f, \text{tot}}}{\mathbf{A}\_{f, \text{ sphere}}} \tag{6}$$

$$\Theta\_{\text{s,total}} = \frac{\mathbf{A}\_{\text{product}} \times \mathbf{n}\_{\text{s,tot}}}{\mathbf{A}\_{\text{s, sphere}}} \tag{7}$$

Here, Aprolate indicates an area projected on the sphere surface as an occupying area. The areas to be covered for the first and the second layer are represented by A*f*,sphere and A*s*,sphere, respectively. The values of A*f*,sphere and A*s*,sphere are different depending the adsorption orientation. The total adsorption point for the first layer (*nf,*tot) and the second layer (*ns,*tot) are illustrated in Figure 14b and detailed in Equation (3) and (4), respectively.

**Figure 14.** (**a**) Sketches of two possible orientation of prolate and corresponding occupying area are shown. Top: spiking out orientation and occupying π*a* <sup>2</sup> on the surface. Bottom: lie-down orientation and occupying π*ab* on the surface. (**b**) For spiking-out orientation, a procedure of counting total number of adsorption points for the first layer (n*f*,tot) and for the second layer (n*s*, tot) are shown. The n*f*,tot is an addition of top (n*f*,top) and bottom (n*f*,bot) adsorption points of a sphere, maximum adsorption position at equatorial positions (n*f*,*eq*), and the adsorption point along axial axis n*f, ax* added along each equatorial position *j* for a semi-sphere. For the second layer, the total number of adsorption points (n*s*,tot) is given by a summation of total number of equatorial adsorption position over a sphere of second layer (n*s*"eq) and total number of adsorption points associated with axial positions, where total adsorption points of each axial line (n*s*,ax) is counted for each equatorial position of second layer, *j*.
