The Study Of Several Models In Nonlinear Mathematical Physics Equations And Systems Biology

This paper studies the traveling wave solutions of several kinds of famous nonlinearmathematical physics equations, that is, classical Drinfel'd-Sokolov-Wilson equation andthe generalization of the Camassa-Holm equation, and the model methods of identifyingphenotype-based responsive modules and causal modules in Systems Biology.On the one hand, by exploiting the qualitative theory of diferential equations and thebifurcation method of dynamical systems, we obtained many new solitary wave solutions,periodic wave solutions, blow-up solutions, periodic blow-up solutions, and kink-shapedsolutions for classical Drinfel'd-Sokolov-Wilson, and new peakons and periodic cusp wavesfor the generalization of the Camassa-Holm equation, and we also discovered new bifur-cation phenomenon. In addition, we have further showed that our qualitative analysis iscorrect by using numerical simulation method. On the other hand, we developed an efec-tive model method to identify phenotype-based responsive modules and causal modules.We exploited the method to study our biological experimental data, i.e., microarray ofbudding yeast cell cycle, and colorectal cancer. We have predicted some new genes relatedto cell cycle and colorectal cancer, and gained some new insight into the mechanism ofthese phenotypes. The main research work are as follows.In Chapter1, we mainly introduce the background, research developments, and basicknowledge of our research objects, and achievements.In Chapter2, by using the qualitative theory of diferential equations and bifurcationmethod of dynamical systems, we study the classical Drinfel'd-Sokolov-Wilson equationand obtain many solutions, which contain diferent kinds of solutions, such as solitarywave solutions, periodic wave solutions, blow-up solutions, periodic blow-up solutions,kink-shaped solutions, and so on. Compared to the previous research of the equation,most of the solutions we obtain are new, which extends previous results in some extent.In Chapter3, we study the peakons and periodic cusp waves for the generalization ofthe Camassa-Holm equation. From the bifurcation theory, in general, the peakons can beobtained by taking the limit of the corresponding periodic cusp waves, when bifurcationparameters tend to special values. However, we found that, even when bifurcation pa-rameters tend to the corresponding special values, the periodic cusp waves will no longerconverge to the peakons, instead, they will still be the periodic cusp waves. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature.In Chapter4, we further extend the peakons and periodic cusp waves for the gener-alization of the Camassa-Holm equation. Compared to the research in Chapter3underspecial parameter conditions, this chapter extends the peakons and periodic cusp wavesfor the generalization of the Camassa-Holm equation to the more general parameter con-ditions. Parts of the results in Chapter3become the special forms of Chapter4.In Chapter5, we develop an approach to identify phenotype-based responsive mod-ules by mathematical programming. We apply the approach to study our experimentaldata, i.e., microarray data of budding yeast cell cycle, and identify important phenotype-and transition-based responsive modules for diferent stages of cell-cycle process. Theresulting responsive modules provide new insight into the regulation mechanisms of cell-cycle process from a network viewpoint. Moreover, the identifcation of transition modulesofers a new way to study dynamical processes at a functional module level. In particular,we found that the dysfunction of a well-known module and two new modules may directlyresult in cell cycle arresting at S phase.In Chapter6, we develop an integrated approach to identify causal module of com-plex diseases by integrating prior information, epigenomic data, gene expression data andprotein-protein interaction network. Compared to the approach in Chapter5, the modelin this chapter is more comprehensive and reasonable, which can be solved more efcientlyand efectively, and the identifed modules are more convincing. We show that the identi-fed modules can indeed serve as efective module biomarkers for characterizing colorectalcancer from diferent perspectives. Further, through constructing TF-module network,we concluded that aberrant DNA methylation of genes encoding TF may contribute toconverted activity of some genes, which may function as causal genes of colorectal cancer,and help develop efcient therapies or efective drugs.In Chapter7, we give a simply introduction of the interdiscipline of mathematicalphysics equations and Systems biology. The study of the interdiscipline help promote thedevelopment and fusion of two disciplines, and have important implications for futureresearch.In Chapter8, we look into the future research. Finally, after the summary of thisdissertation, some problems are proposed for further research and explore.