Research Of Gene Regulatory Network Based On Stochastic Differential Equations Model

Gene regulatory network (GRN) has been a hot topic in bioinformatics in recent years.GRN is a complex network that describes the regulations of the interactions of genes.Through the analysis of GRN, the regulations of the interactions and cooperation patterns canbe revealed, and the functions of the unknown genes can be discovered. One of the importantproblems is the construction of the GRN, and most of the methods are based on the theories ofthe system biology. At this viewpoint, the unknown GRN is regarded as an unknown complexsystem composed by genes, and a proper model is established to describe it. The structure andparameters of this system will be determined by analyzing the relations of the inputs andoutputs of the unknown system. This kind of methods is called reverse engineering or systemidentification, and it is very popular.The ordinary differential equations (ODEs) model is a very popular and typical model forsystem identification. This model uses the ordinary differential equations to depict the kineticregulations of the unknown system, which has a very rigorous mathematical theory supportand can provide a detailed description of the GRN. The ODEs model is convenient forintroducing many mathematical tools to analyze the properties of the network, such as thestability. However, ODEs model is a deterministic model that cannot adjust the intrinsicstochasticity of the biological systems. As a result, the stochastic differential equations (SDEs)model is chosen to complement this pitfall. On one hand, SDEs model can inherit manyadvantages of the ODEs model, on the other hand, SDEs model can cope with the stochasticproblems. The target is to determine the structure and parameters of an unknown SDEssystem by means of the traditional tree-structure based evolutionary algorithms.The SDEs model is an intuitive extension of the ODEs model by appending a stochasticterm to each equation. Unfortunately, this modification introduces stochastic integral andmakes the estimation of the parameters in a SDEs model strikingly difficult. One kind ofmethods are based on mathematical analysis and most of them rely on the solution of theSDEs, and this kind of methods are strict in mathematics but hard to implement automaticallybecause of the necessity of It formula, which can only take effect accompanying a properlyconstructed function. Another kind of methods are based on stochastic simulation, depending on numerous repetitions of numerical solution to estimate the distribution of the sample data.This kind of methods is easy to program but not strict enough in mathematics and extremelytime consuming. Through the comparison and analysis of difference methods, we propose astochastic difference equations (SDCEs) model to eliminate the stochastic integral. It is acompromise between analysis methods and stochastic simulation methods, which can beimplemented by program with efficiency and can avoid complex analysis theories with lessaccuracy loss.The transformation from a SDEs system to its SDCEs system is approximation, and itwon’t be suited for any situation. But it is still sensible concerning with the following reasons:(1) in SDEs model, the troubles caused by stochastic integral are inevitable, but they havelittle relation to the systems identification;(2) the computer can only deal with discrete data;(3) the difference system is common and widely used;(4) in many cases, the differencesystem can approximate its corresponding differential system well by shrinking the samplingperiod (time step);(5) through this transformation, the maximum likelihood estimationmethod can be easily applied and our method can be implemented by program and runs muchfaster than stochastic simulation methods;(6) through this transformation, our method canreveal both structures and parameters of the unknown system, and an solvable prototype ofthe unknown SDEs is not needed any longer. So the SDCEs system will be our focus. Ofcourse, if the original system is a stochastic difference system, this transformation will beskipped and more accurate result can be derived.Then, we transform the SDEs to its corresponding SDCEs to eliminate stochastic integrals,and propose an easy and effective solution for this process. In this solution, the maximumlikelihood estimation can be applied directly and the tree-structure based evolutionaryalgorithms are used to determine structures and parameters of the unknown system.By analyzing the experiments results, some improvements and constraints for this modelare introduced. Strictly speaking, at least two observations of a stochastic system from thesame initial condition can reflect the stochasticity of the system, thus in practice, extraobserved data should be added gradually until the result has no conspicuous improvement.The diffusion coefficient of the unknown system should be relevantly small to drift coefficient.That means the target system should have conspicuous and stable inner laws. If the diffusion coefficient is too big, the algorithm will counterbalance the randomness in the data as apriority. Thus our result will strongly depend on the sample data and whether the algorithmcan find the correct system will become a stochastic event. Because of the stochasticity, thismakes it possible that some complex equations that approach to the sample data well haveeven larger fitness values. Some of the remedies are introduced to tackle this problem, such asto enlarge the effective interval, or to introduce a penalty term to force the expressionssimplified during the evolution. Of course, it is also recommended to apply priori knowledgeto exclude the meaningless solutions and get a better result.For GRN, we point out two bugs of the popular ODEs model: slow reaction and verylimited amount of reactants in biological reactions, which make the ODEs model just aapproximate model. So is the SDEs model. The rigorous disposition of the stochastic integralswon’t bring a significant increase of the accuracy. While the SDCEs suits the biologicalconditions in mathematical theories better than the SDEs model and it is more convenient andefficient than the SDEs model. And in the end, we discuss the advantages of the stochasticdifference equations model, and show the pitfalls of this model. The plans and suggestions ofthe further improvements are also discussed.