The Entropy Characters Of Point Mutation

The life-form is rich, colorful and diversity in nature. The distinction not only exists between the individuals of the different genera but also does between the individuals of the different genera. That is to say a species have the characteristic of diversity. And the characteristic is called by the biological diversity. The biological diversity is a general phenomena and is the foundation that a species evolves from. The entropy is the best method to measure diversity.Some of the biological diversity is due to environmental factors, and some is caused by their own genetic material. The genetic factors are the most important and fundamental decisional factors of the biological diversity. Mutation is the main reason that caused the distinction of genetic material. In order to reveal the influence of mutation on the biological diversity, and eventually show the direction of development of the biological diversity, the characteristics of the Shannon entropy were investigated from the two aspects of theoretical analysis and simulation study using the Shannon information entropy as the method to measure the biological diversity at the molecular level.Theoretical Analysis: First of all, under the mathematical theory about infinity, we discussed the DNA sequences with infinite length and proved that the sequence is always a balance in any mutation rate and the Shannon entropy of the DNA sequence always reaches the maximum entropy ln4; Second, basing on the single-parameter model, we established the differential equations about the point mutation of a DNA sequence with finite length, obtained the functions describing the processes of the variation in quantities of four kinds of bases(A, T, G and C) in the DNA sequence as the generation developed and proved that the Shannon information entropy tends to the maximum entropy as the generation increases; Finally, basing on the two-parameter model, we established the differential equations about the point mutation of a DNA sequence with finite length, obtained the functions describing the processes of the variation in quantities of four kinds of bases(A, T, G and C) in the DNA sequence as the generation developed and proved that the Shannon information entropy also tends to the maximum entropy as the generation increases.Simulation Study: on the basis of theoretical analysis,we did a series of computer simulations of random point mutation in Matlab environment. First of all, several simulations of the entropy of the DNA sequences with the same length and the different mutation rates were done. At first, randomly produced three random DNA sequences with 5×105bp in length using the computer; and then, let them randomly mutate at the average mutation rates of the genome 0.0025/G/generation, 0.0034/G/generation,0.0046/G/generation, respectively; did not differentiate the mutation rates between the four kinds of bases; the simulations passed through 100000, 200000, 500000 generations, respectively; calculated the entropy of every sequence at every generation. And then, Similarly, some simulations of the entropy of the DNA sequences with the same mutation rate and the different length were done too. At first, randomly produced three random DNA sequences with the length 5×103bp,5×105bp,5×106bp, respectively; and then, let them randomly mutate at the average mutation rates of the genome 0.0034/G/generation; did not differentiate the mutation rates between the four kinds of bases; the simulations passed through 100000, 200000, 500000 generations, respectively; calculated the entropy of every sequence at every generation. The two types of simulation results are in line with the theoretical analysis and visually testify the theoretical results.Based on the single-parameter and two-parameter models, We studied the entropy characters of point mutation through theoretical analysis and simulation studies. We found that the entropy of the DNA sequence varies as the generation proceeds and slowly approaches to the maximum entropy, and elucidated the fact that a species evolves in the direction of increasing its diversity.