Symbolic Computation And Its Applications In Computational Biology

The symbolic computation, also known as the computer algebra, is the perfect complement of the traditional logic digestion and numerical calculations, which has been widely used in the nonlinear algebraic equations, the geometry theorem proving, the cryptanalysis, the robotics, the bioinformatics,etc.Academician Wenjun Wu and Academician Jingzhong Zhang proposed Wu’s methods and invariant methods respectively, which have been proved to be the outstanding contribution in the geometric theorem proving. Both methods have laid a solid foundation for the in-depth study of symbolic computation.In recent years, some critical applications problems have massively emerged, with the advent of the era of big data. The symbolic computation gradually shifts from the traditional computer algebra system(CAS) to the mixed symbolic numerical calculations and symbolic logic hybrid reasoning. The research goal is no longer a simple symbolic computation tool provider, but the use of symbolic computation methods and the ideas to solve some significant practical problems.This dissertation has deeply studied Grobner base knot theory and symbolic computation methods, investigated some fundamental problems of artificial intelligence such as algebraic geometry theorem proving methods and logical methods, and researched the major problems in the field of bioinformatics with the methods of numerical symbols. The main work and innovations are presented as follows:I、Dixon resultant has been widely used in solving some complex nonlinear algebraic systems. However, judging KSY conditions needs to build complex theoretical and computational basis. In order to verify KSY conditions, this thesis has proposed a new method, which is based on the elementary transformation matrix and obtain the results after elimination effectively;II 、 Grobner base method is the most effective method to solve the zero-dimensional nonlinear algebraic equations. The latest versions of this method are F4 and F5 algorithms, which have made significant improvements for S-polynomial reduction and greatly improved the efficiency of the Grobner basis method. Based on the different divisions of the polynomial, this dissertation has proposed a calculation method for an extension of S-polynomial, which introduces more terms of a polynomial into the calculation except the first term with the conditon. In comparison with the original S-polynomial, we obtain the polynomial expansion with lower order and higer efficiency of reduction;III、Geometry theorem proving is a basic problem in artificial intelligence, early in 1950, Tarski creatively proposed a mechanical method, but it’s too complex to achieve till now. Fortunately, Wu method effectively solved this problem, but the premise is that each triangular set should satisfy nondegenerate condition. Later, Jinzhong Zhang and Lu Yang proved non-degenerate condition can be changed into weak nondegenerate condition which is verified more easily. On this basis, this dissertation systematically studies the simplifying criterions for triangular sets, thus ensuring the simplified triangular sets not only to meet the nondegenerate condition, but also have a lower order of principal variable. On the other hand, because the human-readable proving process can not be generated by Wu method, plane geometry proving system based on Prolog rools is sdudied in this dissertation, and experiments show the feasibility and scalability of this logical method;IV、Pharmacokinetics is the study of drug absorption, distribution, metabolism and excretion of quantitative rules and disciplines; it applies principle of dynamics in the process. Pharmacokinetics use mathematical formulas to clarify the drug’s relationships among different parts、concentration and time. Its models are of different complexity, one and two-compartment model are popular because of simple mathematical treatment while multi-compartment model is rarely used. A three-compartment pharmacokinetic model of the liver targeting drug DHAQ-PBCA-NP is established by the cooperation with key laboratory of drug targeting and drug delivery System, ministry of education in Sichuan University. Firstly, we determine the morphology and basic properties of the solutions using symbolic computation, then for the first time to solve a three-compartment model based on experienced transport parameters in the country, finally analysis the reliability of the model combined with experimental data;V、The origin of the genetic code is a basic major problem in biology, and there is no accepted academic theory so far. This dissertation creatively proposes a new genetic concept--codon box, based on which the codon table is transferred into the codon box model using bipartite graph theory. The maximal matching number of the codon box model is 19 according to the Hungrian method, and this result can explain the evolutionary theory of the origin of the codon effectively. Moreover,a inference will be given that the codon table evolutes towards to the direction leading to the codon box model to perfect matching. This inference directly explains why the types of amino acids that can synthesize protain are 20. Finally, we can predict that the largest evolutionary probability is GGG coding to Asp rather than the original Gly.In fact, from the perspective of information science, codon table can onlycorrect an error happened on the third position, but the codon box model is able tocorrect the error at any three positions. Furthermore, the greater the number of maximal matching is, the stronger error correction capability the model will has. This nature also proves the inference dissertation proposes.