Long-Range Correlations and Their Measure

For a statistical analysis of copolymer sequences, different mathematical techniques are used. For mathematically oriented researchers, a copolymer sequence might be considered as a string of symbols whose correlation structure can be characterized completely by all possible monomer-monomer correlation functions. Since the correlations at long distances are typically small, it is important to use the best possible estimates to measure the correlations, otherwise the error due to a finite sample size can be as large as the correlation value itself. 

To monitor the long-range statistical properties of computer-generated sequences, the method developed by Stanley and co-workers [43,44] in their search for LRCs in DNA sequences is usually employed. In this approach, each 

The result of calculations [35] averaged over 2000 independent proteinlike HP-sequences of N = 1024 monomeric units with a 1 1 HP composition is presented in Fig. 5. For comparison, the data for two other types of sequences are also shown. One of them is a purely random 1 1 sequence it demonstrates Dx oc A1/2 scaling, as expected. Comparing this curve with 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

To end this section, it is worthwhile to note that long-range dependence processes (also called long-memory processes) and their statistics have many areas of application statistical physics, neuroscience, communication networks, turbulence, hydrology, meteorology, geophysics, finance, econometrics. The literature on the subject is vast (see, e.g., [51]). 

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