## 3 Results
### Working Examples: Fitting Finite Mixture Models to Lifespan Datasets
M is the number of entries to be simulated for the mixture random generation from the model of the finite mixture, p is a positive real number that meets \(0<\text{p}<1\), \(\kappa\) is the number of parameters of a given model, and par is the parameter vector for finite mixture model that carries the same shape as the argument starts of the fitmixEM function. \(\kappa\) is a sole integer value representing the number of unknown

Figure 1: (**a**) Number of daughters (RLS) and frequency histogram of the sir2 deletion genetic background is illustrated. (**b**) The histogram of number of daughters (RLS) of sir2 from a two-component Gompertz model is shown. parameters of a given PDF to contain in the model of the finite mixture. Models of the finite mixture parameters are gauged utilizing the Expectation-Maximization (EM) algorithm. As a demonstration, the script below outputs 500 observations from a two-parameter mixture model of Gompertz and demonstrates the results in Figure 2.

M <- 500 K <- 2 lambda<-c(0.6,0.4) Shape<-c(0.5,1) rate<-c(1,0.5) par<-c(lambda,shape,rate) L <- rmix(M, \"gompertz\", k, par) LL <- seq(0, max(L), 0.01) PDF<- dmix(LL, \"gompertz\", k, par) hist(L, col=\"grey\", breaks=100, ylim=c(0,1.5), las=1, main=\"\") lines(LL, PDF, col=\"blue\")

### Finite Gompertz Mixture Model: Yeast Mutants with Known Effects on RLS Application

To better demonstrate the usefulness of the **fitmix** package, we performed Gompertz mixture model on experimental RLS measurements of _single gene deletion mutants_ with known effects on aging [17; 30]. We estimated parameters of the Gompertz mixture model from RLSs of different yeast mutants shown in Figure 3. Since the Gompertz model is a good fit for lifespan data [9; 31], such as for deletion mutants _tor1_, _sch9_, _sir2_, _hup4_, _sod2_, and _pmr1_, the **fitmix** package is even better because it uses the Gompertz mixture model to portray lifespan data. Figure 2: Histogram plot of 500 randomly distributed realizations from a two-parameter mixture model of Gompertz. In [PERSON] et al. 2020, two subpopulations were reported during replicative aging. In BY4741 wildtype genetic background, cells have two approximately equally weighted subpopulations. In two single-gene deletion mutants, _sir2_-deletion and _hap4_-deletion, the subpopulations are skewed to one subpopulation over another [32]. Our mixture fitting results confirm these significant findings. Based on its histogram, even _hap4_-deletion may have three subpopulations of cells (Table 1 and Figure 4). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Estimated**} & \multicolumn{3}{c}{**Parameters**} \\ \hline deletion mutants & \(\omega_{1}\) & \(\alpha_{1}\) & \(\beta_{1}\) & \(\omega_{2}\) & \(\alpha_{2}\) & \(\beta_{2}\) \\ \hline _tor1_ & 0.58193 & 0.11853 & 0.00025 & 0.41806 & 0.17316 & 0.00118 \\ \hline _sch9_ & 0.32569 & 0.16519 & 0.00037 & 0.67430 & 0.23548 & 0.00188 \\ \hline _sir2_ & 0.41763 & 0.14042 & 0.00215 & 0.58236 & 0.15148 & 0.22705 \\ \hline _hup4_ & 0.32934 & 0.23195 & 0.00099 & 0.67065 & 0.16322 & 0.00031 \\ \hline _sod2_ & 0.49874 & 0.17351 & 0.0001 & 0.50125 & 0.17925 & 0.00353 \\ \hline _pmr1_ & 0.37274 & 0.12604 & 0.00142 & 0.62725 & 0.22176 & 0.00377 \\ \hline wildtypes & \(\omega_{1}\) & \(\alpha_{1}\) & \(\beta_{1}\) & \(\omega_{2}\) & \(\alpha_{2}\) & \(\beta_{2}\) \\ \hline _BY4741_ & 0.57262 & 0.22899 & 0.00139 & 0.42737 & 0.21757 & 0.00007 \\ \hline _BY4742_ & 0.48798 & 0.17110 & 0.00443 & 0.512012 & 0.13422 & 0.00053 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimated parameters of the finite Gompertz mixture model using the **fitmix** package. Figure 3: Overlay of Gompertz mixture fitting curves with RLS histograms in yeast mutants. Red fitting curves represent the Gompertz mixture fit to the RLS data of yeast mutants, and the probability density functions of lifespan data are represented with the blue dashed fitting curves; (**a**) _tor1_, (**b**) _sch9_, (**c**) _sir2_ and (**d**) _hap4_, (**e**) _sod2_, (**f**) _pmr1_.