The aim of the present paper is to investigate the existence of solutions to functionaldifferential inclusions with infinite delay in Banach spaces.A relevant set of phase space axiomsis proposed.The main tools used in this paper are certain fixed point theorems based on the set-contraction theory.
The characterization of long-range correlations and fractal properties of DNA sequences has proved to be a difficult though rewarding task mainly due to the mosaic character of DNA consisting of many patches of various lengths with different nucleotide constitutions. In this paper we investigate statistical correlations among different positions in DNA sequences using the two-dimensional DNA walk. The root-mean-square fluctuation F(l) is described by a power law. The autocorrelation function C(l), which is used to measure the linear dependence and periodicity, exists a power law of C(l) -τμ. We also calculate the mean-square distance ＜R2(l)＞ along the DNA chain, and it may be expressed as ＜R2(l)＞ - l r with 2 ＞γ＞ 1. Our investigations can provide some insights into long-range correlations in DNA sequences.
In protein molecules each residue has a different ability to form contacts. In this paper, we calculated the number of contacts per residue and investigated the distribution of residue-residue contacts from 495 globular protein molecules using Contacts of Structural Units (CSU) software. It was found that the probability P(n) of amino acid residues having n pairs of contacts in all contacts fits Gaussian distribution very well. The distribution function of residue-residue contacts can be expressed as: P(n) = P0 + aexp[-b(n - nc)2]. In our calculation, P0 = -0.06, a = 11.4, b = -0.04 and nc = 9.0. According to distribution function, we found that those hydrophobic (H) residues including Leu, Val, Ile, Met, Phe, Tyr, Cys, and Trp residues have large values of the most probable number of contact nc, and hydrophilic (P) residues including Ala, Gly, Thr,His, Glu, Gln, Asp, Asn, Lys, Ser, Arg, and Pro residues have the small ones. We also compare with Fauchere-Pliska hydrophobicity scale (FPH) and the most probable number of contact nc for 20 amino acid residues, and find that there exists a linear relationship between Fauchere-Pliska hydrophobicity scale (FPH) and the most probable number of contact nc,and it is expressed as: nc = a + b × FPH, here a = 8.87, and b = 1.15. It is important to further explain protein folding and its stability from residue-residue contacts.
