Asymmetric cell division is an important mechanism for the differentiation of cells during embryogenesis and cancer development. Saccharomyces cerevisiae divides asymmetrically and is therefore used as a model system for understanding the mechanisms behind asymmetric cell division. Ace2p is a transcriptional factor in yeast that localizes primarily to the daughter nucleus during cell division. The distribution of Ace2p is visualized using a fusion protein with yellow fluorescent protein (YFP) and confocal microscopy. Systems biology provides a new approach to investigating biological systems through the use of quantitative models. The localization of the transcriptional factor Ace2p in yeast during cell division has been modelled using ordinary differential equations. Herein such modelling has been evaluated. A 2-compartment model for the localization of Ace2p in yeast post-cytokinesis proposed in earlier work was found to be insufficient when new data was included in the model evaluation. Ace2p localization in the dividing yeast cell pair before cytokinesis has been investigated using a similar approach and was found to not explain the data to a significant degree. A 3-compartment model is proposed. The improvement in comparison to the 2-compartment model was statistically significant. Simulations of the 3-compartment model predicts a fast decrease in the amount of Ace2p in the cytosol close to the nucleus during the first seconds after each bleaching of the fluorescence. Experimental investigation of the cytosol close to the nucleus could test if the fast dynamics are present after each bleaching of the fluorescence. The parameters in the model have been estimated using the profile likelihood approach in combination with global optimization with simulated annealing. Confidence intervals for parameters have been found for the 3-compartment model of Ace2p localization post-cytokinesis. In conclusion, the profile likelihood approach has proven a good method of estimating parameters, and the new 3-compartment model allows for reliable parameter estimates in the post-cytokinesis situation. A new Matlab-implementation of the profile likelihood method is appended.

Survival analytic methods help to analyze lifetime data arising from medical and reliability experiments. The popular proportional hazards model, proposed by Cox (1972), is widely used in survival analysis to study the effect of risk factors on lifetimes. An important assumption in regression type analysis is that all relative risk factors should be included in the model. However, not all relative risk factors are observed due to measurement difficulty, inaccessibility, cost considerations, and so on. These unobservable risk factors can be modelled by the so-called frailty model, originally introduced by Vaupel et al. (1979). Furthermore, the frailty model is also applicable to clustered data. Cluster data possesses the feature that observations within the same cluster share similar conditions and environment, which are sometimes difficult to observe. For example, patients from the same family share similar genetics, and patients treated in the same hospital share the same group of profes- sionals and same environmental conditions. These factors are indeed hard to quantify or measure. In addition, this type of similarity introduces correlation among subjects within clusters. In this thesis, a semi-parametric frailty model is proposed to address aforementioned issues. The baseline hazards function is approximated by a piecewise constant function and the frailty distribution is assumed to be a Birnbaum-Saunders distribution. Due to the advancement in modern medical sciences, many diseases are curable, which in turn leads to the need of incorporating cure proportion in the survival model. The frailty model discussed here is further extended to a mixture cure rate frailty model by integrating the frailty model into the mixture cure rate model proposed originally by Boag (1949) and Berkson and Gage (1952). By linking the covariates to the cure proportion through logistic/logit link function and linking observable covariates and unobservable covariates to the lifetime of the uncured population through the frailty model, we obtain a flexible model to study the effect of risk factors on lifetimes. The mixture cure frailty model can be reduced to a mixture cure model if the effect of frailty term is negligible (i.e., the variance of the frailty distribution is close to 0). On the other hand, it also reduces to the usual frailty model if the cure proportion is 0. Therefore, we can use a likelihood ratio test to test whether the reduced model is adequate to model the given data. We assume the baseline hazard to be that of Weibull distribution since Weibull distribution possesses increasing, constant or decreasing hazard rate, and the frailty distribution to be Birnbaum-Saunders distribution. D ́ıaz-Garc ́ıa and Leiva-Sa ́nchez (2005) proposed a new family of life distributions, called generalized Birnbaum-Saunders distribution, including Birnbaum-Saunders distribution as a special case. It allows for various degrees of kurtosis and skewness, and also permits unimodality as well as bimodality. Therefore, integration of a generalized Birnbaum-Saunders distribution as the frailty distribution in the mixture cure frailty model results in a very flexible model. For this general model, parameter estimation is carried out using a marginal likelihood approach. One of the difficulties in the parameter estimation is that the likelihood function is intractable. The current technology in computation enables us to develop a numerical method through Monte Carlo simulation, and in this approach, the likelihood function is approximated by the Monte Carlo method and the maximum likelihood estimates and standard errors of the model parameters are then obtained numerically by maximizing this approximate likelihood function. An EM algorithm is also developed for the Birnbaum-Saunders mixture cure frailty model. The performance of this estimation method is then assessed by simulation studies for each proposed model. Model discriminations is also performed between the Birnbaum-Saunders frailty model and the generalized Birnbaum-Saunders mixture cure frailty model. Some illustrative real life examples are presented to illustrate the models and inferential methods developed here. / Thesis / Doctor of Science (PhD)