5 in the permitted regions of the spectrum f ( α ). This behavior allows an alternative explanation for the dynamics of the clew of a protein, because it suggests the existence of preferential folding trajectories along the en- ergy hyper-surface, i.e. in the search for its native state, proteins need not visit all the accessible states in the space phase, but only those associated with the spectrum f ( α ). associated with the temporal correlation among the dy- namic variables of the series. Our results corroborate some of those obtained by Heg- ger et.al [19] and Lidar and collaborators [22], such as the influence of time and the presence of the α -helix in the rugosity of the time series. However, they also indicate that the other findings have not been confirmed, since the time simulation they used is much shorter than that used in this study and so insufficient to observe the formation of secondary structures. Also, the fractal analysis tech- nique employed by these authors, which does not deal adequately with the existence of trends in the series, has not allowed to capture the subtler details of the spectra. The MF-DFA method applied to the time series of the potential energy of polyalanines, has enabled this study to reveal important aspects concerning the wealth and complexity associated with the temporal evolution of these systems, in the search for its native state. In fact, according to the number of residues and the temperature, it was shown that the trajectory of the protein, to visit its phase space dynamically, is guided mainly by the influence of secondary structures, which are formed over the time simulation, probing the hyper- surface of the conformational energy at different time scales. As a result, the energy time series exhibit multi- fractal long-range correlations. Indeed, the results obtained in this study indicate that all the series examined exhibit typical multifractal be- havior, which depends both on the number of residues N , and the temperature T of the system, and that these multifractal properties, represented by τ ( q ) spectra or, similarly, the generalized Hurst exponents h ( q ), reveal important aspects of the temporal evolution of the sys- tem. Therefore, our results support an alternative explana- tion of the so-called Levinthal paradox [31], because in this scenario, the protein in its dynamic evolution, is be- ing influenced by the emergence of intermediate struc- tures, which gradually, by successive increases in con- formational stability, bypass the trajectories by way of preferential folding. Consequently, the extreme ease with which a protein is folded, despite the huge number of pos- sible configurations, may be attributed to a succession of events, which it experiments, on a multifractal space-time energy hyper-surface. It was found, for example, that when the number of residues approaches the critical number of residues N$_{c}$ , associated with the formation of a significant amount of secondary structures, the temporal correlation regime of the system is changed. In the case N$_{c}$ = 13 and T = 300K, the system is totally anti-correlated, the spectrum τ ( q ) is truly multifractal and rugosity is more pronounced in the region of small fluctuations ( q < 0), as seen in Figure (3a). For other values of N , the results confirm that the two regimes of correlation are present in the series. Recently, Moret and collaborators [30] conducted an analysis of the spectra (profiles) of the potential energy of proteins, in function of the number of dihedral angles φ and ψ , and found that these profiles are described by a real multifractal f ( α ) spectra. They also found that the f ( α ) spectra were sensitive to the number of degrees of freedom of the system, thus illustrating that the dimen- sionality of the phase space influences the accessibility of parts of the hyper-surface of the potential energy, since the proteins adopt conformations in the phase space only This work was partially supported by the Brazilian federal grant agencies CNPq and CAPES, and from FACEPE (Pernambuco state grant agency) under the grants PRONEX EDT 0012-05.03/04 and APQ-0203- 1.05/08). Acknowledgments 168 (1992). [1] S. Havlin, S. V. Buldyrev, A. Bunde, A. L. Goldberger, P. Ch. Ivanov, C.-K. Peng and H. E. Stanley, Physica A 316 , 46 (1999). [2] C. V. Chianca, A. Ticora and T. J. P. Penna, Physica A 357 , 447 (2005). [3] A. Arneodo, E. Bacry, P. V. Graves and J. F. Muzy, Phys. Rev. Lett. 74 , 3293 (1995). [7] P. Ch. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik and H. E. Stan- ley, Nature 399 , 461 (1999). [8] R. L. Costa and G. L. Vasconcelos, Physica A 329 , 321 (2003). [9] Luciano Telesca, Vincenzo Cuomo, Vincenzo Lapenna and Maria Macchiato, Geophysical Research Letters, 28 , 4323 (2001) [10] Luciano Telesca, Vincenzo Cuomo, Vincenzo Lapenna and Maria Macchiato, Chaos, Solitons & Fractals, 21 , 335 (2004). [4] P. Manimaran, Prasanta K. Panigrahi and Jitendra C. Parikh, Phys. Rev. E 72 , 046120 (2005). [5] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and A. L. Goldberger, Phys. Rev. E 49 , 1685 (1994). [11] R. B. Govindan, A. Bunde and S. Havlin, Physica A 318 , 529 (2003) [6] C.-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Sciortino, M. Simons and H. E. Stanley, Nature 356 ,