The Construction And Preliminary Analysis On Differential Equation Model Of Genetic Regulatory Network

Along with the development of Bioinformatics, based on differential single-gene expression analysis and clusters of genes in terms of common functionality, it has been stringent necessity and hot research problem to infer and understand underlying pattern and mechanism through studying interaction between genes or proteins as a whole and constructing gene regulatory network and protein regulatory network. At present, several models have been applied to model genetic regulatory network: Directed graph, Boolean network, Bayesian network, Differential equation, Stochastic equation and so on.Differential equation model is classified as ordinary differential equation, partial differential equation and stochastic differential equation. We must select a kind of formalism from a number of possible alternatives to analyze interaction between genes during modeling regulatory network, the first method, which can be employed to choose structure of model, is to draw scatter plot of target gene and its every possible regulator, and make a decision by scatter plot; second, to estimate parameters of several possible models, and the model, which produces the least MSE or the most R 2, is selected. In this study, we employ sigmoid function as regulatory function because of its favorable mathematical characteristic.One of the most key puzzles is that the amounts of genes are much more than the numbers of measured times, so it is impossible to estimate effectively parameters. Because the regulators of per gene are finite according to biology, in this paper we identify the regulators of determined target genes, and then construct equations for target genes. Correlation coefficient may evaluate interaction between genes, however, it can be only employed in certain cases. We propose polynomial regression to measure the nonlinear relation of genes. To perform polynomial regression for the target gene and every gene in system, the less is RMSE , the more possible regulator is gene. Finally, considering the results both correlation analysis and polynomial regression, we select several genes as regulators for each target gene, so dimensions are reduced.Estimation of parameters may use classical iterative methods, but these methods are likely to converge to a local minimum, even the iterative process may diverge; we also use recent rising genetic algorithm to estimate parameters, similarly, prematurity phenomenon exists in genetic algorithm, that is quickly converge to a local rather than a global minimum. We combine genetic algorithm with classical iterative methods, after using independently Gauss-Newton method and genetic algorithm to infer parameters, in the first place, the estimation values of genetic algorithm are considered as initial parameter values to supply for Gauss-Newton method, then to perform iterative progress again, observe if the values is better or the iterative process may converge; in the next place, to fix fitness of genetic algorithm and end condition by means of SSE derived from Gauss-Newton method, run genetic algorithm once again to improve results.The aim to construct regulatory network is to find solutions of model and analyze character of system. For linear system, information of regulatory network lies in coefficient matrix, and common method can be employed to solve equations. While in most cases, it is hardly possible to work out analytical solutions for nonlinear system, universal method is the numerical solution. Stability is principal problem of dynamic system theory, meanwhile, is one of qualitative theory of differential equation. Stability of system is studied chiefly according to stability theory proposed byА.М.Ляпунов.In our research, gene expression data of Saccharomyces cerevisiae is applied to construct regulatory network. To choose ten genes as target genes, and then select five genes as regulators for each target gene. We estimate parameters according to our methods. Compared with using independently Gauss-Newton method or genetic algorithm, we find that the combination of two methods optimize results to a great extent, it indicate different methods may be employed together.Our study discuss the construction of regulatory network, solution and stability of system, provide useful thought and idea, gain some meaningful results, form a base for forthcoming research. Certainly, we must cooperate with biologists and study further some questions to play an important role in practical application.