The power spectrum of series with correlated terms: 
a model for protein structure and mobility

C. Vamos1, A. Pop2, V.V. Morariu3

1Computation Institute “Tiberiu Popoviciu” Cluj-Napoca
2Astronomical Observatory, Babeş-Bolyai University, Cluj-Napoca
2INCDTIM, POBox 700, Cluj-Napoca 5, 400293

1/f noise appears in nature all over the places. Its general form is 1/fβ where 0<β<2. It was initially thought that such a noise is the result of a universal mechanism. In the meanwhile it was realised that there exists a wide diversity of mechanisms generating 1/f noise. Mandelbrot’s repeated statements that “…I think that a unique physical explanation for all 1/f noises is hopless…and…that the identification of a 1/f spectrum is never enough.”, has often been  overlooked. One of the disquieting features of the generally so called 1/f spectrum is that very often it is not a “pure” 1/f spectrum (i.e. the log-log plot of the spectrum is not linear over any number of scales). Instead, the log-log plot of the spectrum is linear over a limited range of frequencies while at lower and higher frequencies there are significant deviations from linearity. However the reports in the literature generally ignore such features and the authors are content to approximate the whole spectrum with a “pure” 1/f spectrum (i.e. a linear approximation of the log-log plot of the spectrum). A”pure” 1/f spectrum for a natural process is characteristic, for example, for the Brownian motion. On the other hand we can cite various cases where significant deviations from a 1/f spectrum is present: astrophysical phenomena, laminar flow, river flow, protein atomic mobility, potential energy of proteins, wood fractured surfaces, human activity, cognitive phenomena.
In an earlier publication we proposed a simple correlation model which generated spectra, similar to a first approximation to 1/fβ fluctuations.
However it was quite evident that deviations from a “pure” 1/f spectrum were present in the spectra of the series generated by the model yet this aspect was not investigated. We returned to such a model as it appeared to have a significant potential to describe many cases with deviations from “pure” 1/fβ spectra. 
The purpose of the present work is to systematically describe the statistical properties of the simplest of a correlation model, where interaction exists between the neigbours in the series. We found that the theoretical spectrum signficantly deviates from a “pure” 1/f spectrum. Then we quantiatively analyse the deviation of such a model from the real series of the atomic mobility of the protein main chains. We found that some proteins can be well described by the simplest correlation model while others require additional correlations. We believe this simple model can be applied to many other processes while more ellaborated variants of the model can be applied to more complex phenomena. Further we regard the present work as an attempt to go beyond the simple reporting of an 1/f spectrum and, in fact, to describe a class of phenomena, basically a system with local interaction among the terms, which previously were approximated with the 1/fβ noise.