We investigate the chaotic behaviour of multiparticle systems, in particular DNA and graphene models, by applying methods of nonlinear dynamics. Using symplectic integration techniques, we present an extensive analysis of chaos in the Peyrard-Bishop-
Dauxois (PBD) model of DNA. The chaoticity is quantified by the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair (BP) disorder on the dynamics is studied. In addition to heterogeneity due to the ratio of adenine-thymine (AT) and guanine-cytosine (GC) BPs, the distribution of BPs in the sequence is analysed by introducing the alternation index $alpha$. An exact probability distribution for BP arrangements and $alpha$ is derived using Polya counting. The value of the mLE depends on the composition and arrangement of BPs in the strand, with a dependence on temperature. We probe regions of strong chaoticity using the deviation vector distribution, studying links between strongly nonlinear behaviour and the formation of bubbles. Randomly generated sequences and biological promoters are both studied. Further, properties of bubbles are analysed through molecular dynamics simulations. The distributions of bubble lifetimes and lengths are obtained, fitted with analytical expressions, and a physically justified threshold for considering a BP to be open is successfully implemented. In addition to DNA, we present analysis of the dynamical stability of a planar model of graphene, studying the mLE in bulk graphene as well as in graphene nanoribbons (GNRs). The stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both armchair and zigzag edge GNRs, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.
We investigate the distribution of bubble lifetimes and bubble lengths in DNA at physiological temperature, by performing extensive molecular dynamics simulations with the Peyrard-Bishop-Dauxois (PBD) model, as well as an extended version (ePBD) havi
ng a sequence-dependent stacking interaction, emphasizing the effect of the sequences guanine-cytosine (GC)/adenine-thymine (AT) content on these distributions. For both models we find that base pair-dependent (GC vs AT) thresholds for considering complementary nucleotides to be separated are able to reproduce the observed dependence of the melting temperature on the GC content of the DNA sequence. Using these thresholds for base pair openings, we obtain bubble lifetime distributions for bubbles of lengths up to ten base pairs as the GC content of the sequences is varied, which are accurately fitted with stretched exponential functions. We find that for both models the average bubble lifetime decreases with increasing either the bubble length or the GC content. In addition, the obtained bubble length distributions are also fitted by appropriate stretched exponential functions and our results show that short bubbles have similar likelihoods for any GC content, but longer ones are substantially more likely to occur in AT-rich sequences. We also show that the ePBD model permits more, longer-lived, bubbles than the PBD system.
We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for
a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density, and that the onset of chaos in graphene is slow, becoming evident after more than $10^4$ natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges the chaoticity of GNRs is stronger than the periodic graphene sheet, and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Further, we show that the composition of ${}^{12}C$ and ${}^{13}C$ carbon isotopes in graphene has a minor impact on its chaotic strength.
We discuss the effect of heterogeneity on the chaotic properties of the Peyrard-Bishop-Dauxois nonlinear model of DNA. Results are presented for the maximum Lyapunov exponent and the deviation vector distribution. Different compositions of adenine-th
ymine (AT) and guanine-cytosine (GC) base pairs are examined for various energies up to the melting point of the corresponding sequence. We also consider the effect of the alternation index, which measures the heterogeneity of the DNA chain through the number of alternations between different types (AT or GC) of base pairs, on the chaotic behavior of the system. Biological gene promoter sequences have been also investigated, showing no distinct behavior of the maximum Lyapunov exponent.
In modeling DNA chains, the number of alternations between Adenine-Thymine (AT) and Guanine-Cytosine (GC) base pairs can be considered as a measure of the heterogeneity of the chain, which in turn could affect its dynamics. A probability distribution
function of the number of these alternations is derived for circular or periodic DNA. Since there are several symmetries to account for in the periodic chain, necklace counting methods are used. In particular, Polyas Enumeration Theorem is extended for the case of a group action that preserves partitioned necklaces. This, along with the treatment of generating functions as formal power series, allows for the direct calculation of the number of possible necklaces with a given number of AT base pairs, GC base pairs and alternations. The theoretically obtained probability distribution functions of the number of alternations are accurately reproduced by Monte Carlo simulations and fitted by Gaussians. The effect of the number of base pairs on the characteristics of these distributions is also discussed, as well as the effect of the ratios of the numbers of AT and GC base pairs.
