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correlations, then for increasing values of s , function F$_{q}$ ( s ) also grows, following a power law of the type: 

Therefore, the main result obtained with the MF-DFA method is a family of exponents h ( q ), called the gener- alized Hurst exponents. For a genuine multifractal series these exponents form a decreasing function of q , if on the other hand, the signal is monofractal h ( q ) = constant. Moreover, if q < 0, h ( q ) captures the properties of small fluctuations, then for q > 0 large fluctuations are domi- nant. Particularly, when q = 2, h (2) = H is the classical Hurst exponent. 

Finally, the multifractal spectrum of measures can be obtained through a simple relation between the exponent h ( q ) and the multifractal scale exponent τ ( q ), defined by multifractal formalism [15]: 

The function τ ( q ) is one of the most used representations of multifractal spectra, related to time series. 

Furthermore, typical results are presented obtained us- ing the MF-DFA technique to investigate the different time series of the potential energy of polyalanines as de- scribed in Section II. Figure (2) represents the behavior of the logarithmic of the fluctuation function log F$_{q}$ ( s ) as a function of log s and the parameter q , for the series with N = 18 residues shown in Figure (1). The scale values were chosen in the range 20 < s < 100 and the trends were approximated by a polynomial of order m = 4. As can be observed, the estimates obtained for the linear adjustment of the data satisfactorily meet the behavior of the scale provided by Equation (6). 

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Figure (3a) presents the corresponding exponents h ( q ) (values of the slopes of the straight lines fitted to the 

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data) as a function of q , while Figure (3b) presents the associated multifractal spectrum τ ( q ). In general terms, it may be stated that the results indicate that the time series investigated exhibit typical multifractal behavior ( τ ( q ) is not a linear function of q ), which depends on the number of residues N and the thermal temperature T of system. 

In Figure (3) different regimes of correlation may be observed: for N = 17, the series is completely correlated; while for N = 10, 15 and 18, there is a mixed system, i.e. a strong anti-correlation when q > 0 and correlation, for some values of q < 0. In particular, when N = 13 the series is totally anti-correlated. According with reference [27] N = 13 is the critical number of residues associated with the formation of α -helix in T = 300K. 

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Since for this value of N , the potential energy is a global minimum then we may consider that the nucle- ation of secondary structures alters the dynamics of the system for an anti-correlated regime, thus overcoming the growth trend of the energy induced by thermal ag- itation and the increase of residues. In addition, as in Figure (3b), all spectra τ ( q ) exhibit typical multifractal properties. 

In this work, we have studied the multifractal proper- ties of time series of the potential energy of polyalanines. Protein chains were analyzed with different numbers of residues at three equilibrium temperatures. The research was conducted using an approach that combines molec- ular dynamics with MF-DFA, a technique of statistical analysis, which enabled us to characterize the rugosity 

## IV. CONCLUSIONS 

