# Synthetic Data Generation for the Development of 2D Gel Electrophoresis Protein Spot Models
## 2 Materials and Methods
### Generation of Synthetic Protein Spot Images
#### 2.1.3 Generation of Protein Spot 2D Prototype (Primary Spot)
Edge templates are generated using mathematical functions (Equations (1)-(6)), which are described below. Plots of the functions are shown in Figure 6. Edge templates are generated by randomly selecting a function from the set of six functions (Equations (1)-(6)) and by randomly sampling parameters from the defined range (for parameter ranges, refer to Table 1). Figure 6: Visualizations of the functions used to create 1D protein spot edge templates. The graph shows the plots of the functions: Gaussian (**a**), simplified diffusion process (**b**), spline (**c**), Fuzzy Logic generalized membership bell-shaped function (**d**), sigmoid (**e**), raised cosine (**f**). The horizontal axis spans the distance from the spot center to the edge of the image sample (in this research, \(M=28\) px, half of the image patch size, which is \(56\times 56\) px). The abilities of the functions to imitate the shapes of protein spots are revealed by random sampling parameters from the defined range and plotted as thin lines. Thick lines represent the cases when parameters are being set to the minimum or maximum of the defined range (refer to Table 1). Functions that determine the profile of the protein spot edge template:  
1. **Gaussian** function with its center at \(x=0\) (the graph of the function is shown in Figure 5(a)): \[f_{\text{G}}(x;\sigma,d)= \begin{cases}1,&\text{if $x\leq d$;}\\ e^{\frac{-(x-a)^{2}}{2\sigma^{2}}},&\text{otherwise,}\end{cases} \] (1) here \(d\) --shift of the mean of the Gaussian function from the spot center (\(x=0\)); \(\sigma\)--standard deviation of the Gaussian function. 2. Simplified **diffusion** process function with its center at \(x=0\) (refer to Figure 5(b)): \[f_{\text{D}}(x;D,a)=\frac{1}{2}\Big{[}\text{erf}\Big{(}\frac{a+r}{2}\Big{)}+ \text{erf}\Big{(}\frac{a-r}{2}\Big{)}\Big{]}+\frac{1}{r\sqrt{\pi}}\Big{[}e^{- \big{(}\frac{a+r}{2}\big{)}^{2}}-e^{-\big{(}\frac{a-r}{2}\big{)}^{2}}\Big{]},\] (2) here \(r=|x|/\sqrt{D}\); \(D\)--parameter defining the diffusion (defines a slope and a shift from the center of the curve transition area); \(a\) is the radius of diffusion area. 3. **Spline** function with its center at \(x=0\) (refer to Figure 5(c)): \[f_{\text{Sp}}(x;a,b)= \begin{cases}1,&\text{if $x<a$;}\\ 1-2\big{(}\frac{x-a}{b}\big{)}^{2},&\text{if $a\leq x<\frac{a+b}{2}$;}\\ 2\Big{(}\frac{x-a-b}{b}\Big{)}^{2},&\text{if $\frac{a+b}{2}\leq x<a+b$;}\\ 0,&\text{if $x\geq a+b$,}\end{cases} \] (3) here \(a\) is the distance from the center to the beginning of the curve transition area; \(b\) is the distance from the \(a\) to the end of the curve transition area. 4. **Fuzzy Logic** generalized membership **bell-shaped** function with its center at \(x=0\) (refer to Figure 5(d)): \[f_{\text{B}}(x;a,b)=\frac{1}{1+|\frac{x}{a}|^{2b}},\] (4) here \(a\) determines the width of the function's transitional area; \(b\) defines the shape of the curve (slope) where a larger value gives a steeper function. 5. **Sigmoid** function with its center at \(x=0\) (Figure 5(e)): \[f_{\text{Sp}}(x;a,d)=\frac{1}{1+e^{a(x-d)}},\] (5) here \(a\) defines the slope of the transition area of the curve; \(d\)--distance from the center to the curve transition area. 6. **Raised cosine** function with its center at \(x=0\) (Figure 5(f)): \[f_{\text{C}}(x;a,d)= \begin{cases}1,&\text{if $x<d$;}\\ \frac{1}{2}\Big{(}1+\cos(\frac{x-d}{a}\pi)\Big{)},&\text{if $d\leq x<d+a$;}\\ 0,&\text{otherwise,}\end{cases} \] (6) here \(a\) is the width of the transition area of the curve; \(d\)--distance from the center to the curve transition area. Function Parameter Ranges for Edge Template Generation  
Table 1 summarizes the parameter value ranges used to generate 1D protein spot edge templates. The ranges of the parameters are adjusted for the generation of edge templates of length \(M=28\) px (half of the image patch size, which is \(56\times 56\) px in the current research).