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Specifically, we simulate polyalanine structures with a different number of residues at different equilibrium temperatures and initial conformations. Polyalanines are used as prototypes to study the folding process of struc- tures in α -helix conformations. In this dynamic, the elec- tric dipole moments arising from the electric unbalance between the peptide bond of NH and CO groups, the hy- drogen bridges bonds and the van der Waals interactions, are key ingredients in the cooperative effect responsible for forming such structures, and which becomes acceler- ated with the increasing number of amino acids in the protein. 

Thus, as pointed out by Shoemaker and collabora- tors [26], Moret and collaborators [27] and Rogers [28], a critical minimum number of amino acids is necessary so that these configurations may be observed. Furthermore, there is an upper critical number due to destabilization brought on by entropic effects. 

For the numerical calculations, a similar protocol was adopted in all cases. The initial temperature started at T$_{i}$ = 1K, heating the system at a rate of 5K per step (ps) in order to reach the desired equilibrium temperature. Three equilibrium temperatures were considered: T = 275K, T = 300K and T = 325K, in a continuous medium, described by a relative dielectric constant ϵ$_{r}$ = 2. The increases in the time dynamic was 5 10 − 4 ps, and for all simulations N$_{step}$ = 5 10 8 steps were performed, to achieve a time of the order of 25ns. In calculating the time, the interval associated with the thermalization of the system was discarded. 

Figure (1) shows in the last 5ns of observation for the potential energy time series associated to polyala- nine structures in T = 300K and N = 10, 15, and 18. It was noted that in all cases examined, the series showed the typical rugosity observed in other complex phenom- ena described by self-affine time series [13]. It should 

be emphasized that for all temperature values, calcula- t ions were reached with values of N between 8 and 18 re sidues, the results of which display similar behavior to th ose presented in Figure (1). 

## III. MF-DFA – MULTIFRACTAL SPECTRA 

Once the time series of polyalanines have been ob- ta ined and the presence of rugosity has been observed, a careful characterization of the statistical fluctuations e mbedded in the series should be performed in order to obtain information on the dynamic behavior of the sys- te m. In this work, the MF-DFA method is applied, along th e following steps [15]: 

1. Consider a time series u ( i ), i = { 1 ...N$_{max}$ } , over a compact support, and determine its profile (inte- grated set), i.e., 

where < u > is the mean taken over the original series u ( i ); 

2. Divide the profile in N$_{s}$ disjointed segments of equal size s , calculate the local trend through a poly- nomial adjustment of order m , represented by the variable y$_{ν}$ ( i ), at each segment. Since the length N of the series is often not a multiple of the considered time scale s , a short part at the end of the profile may remain. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Thereby, 2 N$_{s}$ segments are obtained altogether; 

3. Determine the fluctuation variance, 

with ν = { 1 , ..., N$_{s}$ } , associated to each segment. Notice that in this step, polynomial trends of the order m were eliminated from the profile. 

4. Calculate the mean values of all segments, to obtain the fluctuation function of the order q : 

where, in general, the variable q assumes real val- ues, except zero. 

The characteristic property of function F$_{q}$ ( s ) is its scale behavior, i.e. if the time series u ( i ) possess long-range 

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