International Center for Mathematical Modelling in Physics and Cognitive Sciences, University of Vaxjo, Vaxjo, S-35195, Sweden.
J Theor Biol. 2009 Dec 7;261(3):396-406. doi: 10.1016/j.jtbi.2009.08.014. Epub 2009 Aug 20.
In this paper we demonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, degeneracy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic distance is an ultrametric and applications of ultrametric in bioinformatics are not surprising. However, by using the 2-adic numbers we match ultrametric with a number theoretic structure. In this way we find new applications of an ultrametric which differ from known up to now in bioinformatics. We obtain the following results. We show that the PAM matrix A allows the expansion into the sum of the two matrices A=A((2))+A((infinity)), where the matrix A((2)) is 2-adically regular (i.e. matrix elements of this matrix are close to locally constant with respect to the discussed earlier by the authors 2-adic parametrization of the genetic code), and the matrix A((infinity)) is sparse. We discuss the structure of the matrix A((infinity)) in relation to the side chain properties of the corresponding amino acids. We introduce the family of substitution matrices A(alpha,beta)=alpha A((2))+beta A((infinity)), alpha,beta>or=0 which should allow to vary the alignment procedure in order to take into account the different chemical and geometric properties of the amino acids.
在本文中，我们证明了使用 2-adic 数系统为遗传学中的一些问题提供了新的见解，特别是遗传密码的简并性和生物信息学中 PAM 矩阵的结构。2-adic 距离是超度量的，超度量在生物信息学中的应用并不奇怪。然而，通过使用 2-adic 数，我们将超度量与一个数论结构相匹配。通过这种方式，我们找到了超度量在生物信息学中不同于已知应用的新应用。我们得到了以下结果。我们表明，PAM 矩阵 A 可以扩展为两个矩阵的和 A=A((2))+A((infinity))，其中矩阵 A((2))是 2-adic 正则的（即该矩阵的元素相对于作者之前讨论的遗传密码的 2-adic 参数化是接近局部常数的），并且矩阵 A((infinity))是稀疏的。我们讨论了矩阵 A((infinity))的结构与相应氨基酸的侧链性质之间的关系。我们引入了替代矩阵族 A(alpha,beta)=alpha A((2))+beta A((infinity))，alpha,beta>=0，这应该允许改变对齐过程，以考虑到氨基酸的不同化学和几何性质。
