2 number of amino acids N . Section III presents the mul- tifractal spectra, obtained from the MF-DFA technique, associated to each different polyalanine time series. The effects caused by changes on the size of the chain and temperature and the presence of secondary structures on the spectra are discussed. Finally, Section IV presents our conclusions. One highly relevant problem in molecular biology within this context, is linked to studies concerning the protein folding process through the characterization of its potential energy landscape. Such landscapes consti- tute a satisfactory representation of the potential energy for interaction among the various system’s microscopic freedom degrees [19]. In general, the adopted strategies are based on the as- sumption that the energy landscapes of proteins are com- plex, since being time dependent, they present a rugose structure, with many maxima and minima separated by barriers of varying heights. These properties imply com- plex evolutionary dynamics, in which the system experi- ences a variety of time scales [20, 21]. Previous studies, using molecular dynamics simula- tions (MOIL program) and a variational method of frac- tal analysis to study the fractal properties of time series of the potential energy of molecular systems such as myo- globin; polyalanines, among others, were conducted by Lidar and collaborators [22]. Basically, they investigated systems that were subjected to a temperature T = 300K and a simulation time in the range 10 < t < 25 ps. Molecular dynamics simulations have been extensively used to study the problem of protein folding [23]. In general, these simulations involve considerable compu- tational effort, since the integration of the equations of motion must be made for a system with many particles. In the case of molecular systems, it is known that such structures can take on a great number of configurations, which grow with the number of degrees of freedom of the system. Therefore, molecular dynamics calculations for protein systems necessarily require the definition of effective potentials, from which the resulting force that acts on each particle is determined. In this work, the numerical molecular dynamics calcu- lations were performed with the aid of an efficient com- puter code: the THOR program [25], developed to inves- tigate structures of biological interest, such as proteins. The code includes the GROMOS force field [24] in its architecture, used to simulate the atomic interactions in the molecule. Their results suggest that the value of the fractal di- mension (the rugosity exponent) slightly depends on tem- perature and the presence of α -helix structures smoothes the rugosity of the series. Furthermore, there was evi- dence of universal behavior, i.e. the rugosity of different systems is described by the same fractal dimension. Re- cently, Hegger et al. [19] analyzed time series extracted from molecular dynamics simulations (GROMACS pro- gram) at a temeperature of T = 300K, for polyalanines with the number N of amino acids ranging between 3 and 10, reaching simulation time of 100 ns. II. MOLECULAR DYNAMICS AND TIME SERIES In the THOR program, the conformational energy of the molecule is made up of a sum of bonded and non- bonded terms. In such approach, only hydrogen atoms covalently bonded to oxygen or to nitrogen are consid- ered explicitly, whereas CH1, CH2, and CH3 groups are assumed to be an atomic unit. Thus, we analyze the changes of the following energy function: Considering that this time series represent the dynamic trajectories followed by the system, these authors found that the effective fractal-dimension of such trajectories decreases with the chain size. According to them, such behavior occurs due to a stabilizing effect of the hydro- gen bonds on the protein secondary structure ( α -helix) smoothing rugosities on the trajectories. Confirming whether this scenario is able to survive careful fractal analysis, searching for fine details of the time series fluc- tuations, has become a central problem to be clarified. The present work introduces an approach, which com- bines molecular dynamics simulations with MF-DFA to characterize the rugosity of potential energy profiles, for polyalanine molecules. By considering these profiles as energy time series, we investigate the effects produced on the trajectories traced over the hyper-surface of the potential energy, when the size and temperature of the system is changed. In particular, we will show that the manner in which the system visits the phase-space in its dynamic evolution significantly depends on both temper- ature equilibrium and the nucleation of secondary struc- tures in polyalanines. This article is organized as follows: Section II presents the molecular dynamics simulations, the energy time se- ries, and the energy dependence on temperature T and where E$_{H}$ is the Hook potential, E$_{θ}$ is the angular po- tential, E$_{φ}$ is the umproper potential, E$_{ϕ}$ is the dihedral- angle potential, E$_{LJ}$ is the Lennard-Jones potential, and E$_{el}$ is the Coulomb potential term (see definitions and used parameters in [24, 25]). or explicitly, E = E$_{H}$ + E$_{θ}$ + E$_{φ}$ + E$_{ϕ}$ + E$_{LJ}$ + E$_{el}$ (1) E = 1 2 ∑ k K$_{b}$$_{k}$ ( r$_{k}$ + r$_{0}$ ) 2 + 1 2 ∑ l K$_{θ}$$_{l}$ ( θ$_{l}$ + θ$_{0}$ ) 2 + + 1 2 ∑ m K$_{φ}$$_{m}$ ( φ$_{m}$ + φ$_{0}$ ) 2 + + n K$_{ϕ}$$_{n}$ (1 + cos( mϕ$_{n}$ + ϕ$_{0}$ )) + (2) + ∑ i<j [ C$_{12}$ ( i, j ) r 12 i.j − C$_{6}$ ( i, j ) r 6 i,j ] + 1 4 πε$_{0}$ε$_{r}$ ∑ i<j q$_{i}$q$_{j}$ r$_{i,j}$ .