Multifractal Analysis of Polyalanines Time Series P. H. Figueirˆ edo, 1, ∗ E. Nogueira Jr., 2, † M. A. Moret, 3, ‡ and S´ ergio Coutinho 4, § $^{1}$Departamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900, Recife, Pernambuco, Brazil $^{2}$Instituto de F´ısica, Universidade Federal da Bahia 40130-240, Salvador, Bahia, Brazil. $^{3}$Departamento de F´ısica, Universidade Estadual de Feira de Santana, 44031-460, Feira de Santana, Bahia, Brazil. $^{4}$Laborat´ orio de F´ısica Te´ orica e Computacional, Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife, Pernambuco, Brazil (Dated: October 29, 2018) Multifractal properties of the energy time series of short α -helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique ( multifractal detrended fluctu- ation analysis ). Estimates for the generalized Hurst exponent h ( q ) and its associated multifractal exponents τ ( q ) are obtained for several series generated by numerical simulations of molecular dy- namics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects on the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process. PACS numbers: 87.15.Cc, 87.15.-v, 05.45.Tp, 64.60.al Keywords: Time series, Self-affinity, Protein folding, Energy landscape these studies stems from its easy implementation. More- over, it is a tool that allows the role of trends in stationary time series to be analyzed, as well as efficiently estimat- ing the long-range correlations through a single param- eter: the scale exponent α . Is important to emphasize that the type of correlation present in the stationary se- ries depends on the value found for exponent α . In this way, for α = 0 . 5 the signal is uncorrelated (white noise or Gaussian), while for α < 0 . 5 there is anti-correlation (anti-persistence) and for α > 0 . 5 there is correlation (persistence) [13]. Over the past few years, the statistical analysis of self- similar time series has become established as an impor- tant tool for investigating several natural phenomena. In general, a large part of these studies has been devoted to characterizing the complex statistical fluctuations shown by these series. Such fluctuations are associated to long- range correlations among the dynamic variables present in these series, and which obey the behavior usually de- scribed by fractal power-law decay [1]. One of the difficulties encountered in these investiga- tions is related to the fact that the series may contain het- erogeneous properties imposing certain statistical trends over itself. In other words, these series are not station- ary [2]. Therefore, it is necessary to employ a technique capable of accounting, for this, since these trends may influence the correlations that exist in the series. Several attempts to apply the DFA technique to non- stationary time series (series affected by local trends) have not provided satisfactory results. Fundamentally, this has occurred because these series are not entirely characterized by a unique scale exponent α , since differ- ent segments possess fluctuations characterized by dis- tinct values of α . In this case, the correct formalism for obtaining the distribution of scale exponents is multifrac- tal analysis [13]. Two techniques have proved successful in eliminating these trends in time series: the wavelet transform mod- ulus maxima (WTMM) [3, 4] and the detrended fluctu- ation analysis (DFA) [5]. Both techniques are based on local polynomial regression in order to eliminate local trends present in different segments of the series. The DFA technique has been particularly efficient for a large range of areas such as: DNA sequences [6]; heartbeat analysis [7]; economy [8]; seismology [9, 10]; meteorology [11]; astrophysics [12], among others. Basically, the option of applying the DFA technique to Recently, the number of works focusing on the char- acteristics of the multifractal aspects of non-stationary time series has grown, particularly those based on ex- perimental data [14]. Outstanding among these are the applications of the generalized DFA technique, known as Multifractal Detrended Fluctuation Analysis (MF-DFA) as proposed by Kanterlhardt and collaborators [15], for a wide range of applications such as: DNA sequences [16], meteorology [17], seismology [18] and others. It should also be remembered that two factors influence the use of the MF-DFA technique: its effortless implementation, and its excellent performance in obtaining results, re- lated to both artificial and real data, when compared to the performance of the Wavelet Transform process, for the same systems [14]. Typeset by REVT E X $^{∗}$Electronic address: phugof@df.ufrpe.br $^{†}$Electronic address: enogue@fis.ufba.br $^{§}$Electronic address: sergio@ufpe.br $^{‡}$Electronic address: moret@cairu.br arXiv:1001.1999v1 [physics.bio-ph] 12 Jan 2010 I. INTRODUCTION