Nonlinear Analysis Of Biological Sequences

In this paper, we mainly investigate nonlinear dynamics in DNA sequence using nonlinear theory methods. That is to say, by studying the correlation between the similar segments of DNA sequence, we try to find the nonlinear dynamics in DNA sequence. The purpose of this paper is to answer the basic problem of biological sequences, e.g.'if DNA sequence is random or determined'; the result of this paper has a certain value to basic research of Bioinformatics.Firstly, we describe the graphical representation of biological sequence, and introduce the CGR map, which is based on iterated function system. CGR produces a picture of a gene sequence which displays both local and global patterns; the picture has a complex structure that varies depending on the sequence. CGR is an important tool for investigate gene structure, and also it is one prerequisite for our further study on nonlinear dynamics in DNA sequence.Secondly, to describe the fractal feature of genomes sequences, a multifractal theory is presented in the analysis. We select some long gene sequences and these sequences are translated into time series according to their coding/noncoding segments, then the Hurst exponent of the time series is calculated and their self-comparability are further analyzed according their Hurst exponent. Comparing Hurst exponent with DNA walk model, it is found that these gene sequences have long-range correlation.Finally, we introduce nonlinear time series methods, and give the definition and calculation formula of correlation dimension, Kolmogorov entropy, and lyapunov exponent, that is the prerequisite for analyzing the chaos process of DNA sequence. We investigate the long memory process of DNA sequence according their Hurst exponent, and then describe the chaos process of the DNA sequences by analyzing their correlation dimension, entropy and Lyapunov exponents. The result is that there is an indication of the existence of chaos in the DNA sequences, and there is a certain relationship between long memory and chaos of DNA sequences.