Periodic, quasiperiodic, fractal, Kolakoski, and random binary polymers: Energy structure and carrier transport

We study periodic, quasiperiodic (Thue-Morse, Fibonacci, period doubling, Rudin-Shapiro), fractal (Cantor, generalized Cantor), Kolakoski, and random binary sequences using a tight-binding wire model, where a site is a monomer (e.g., in DNA, a base pair). We use B-DNA as our prototype system. All sequences have purines, guanine (G) or adenine (A), on the same strand, i.e., our prototype binary alphabet is $\{\mathrm{G},\mathrm{A}\}$. Our aim is to examine the influence of sequence intricacy and magnitude of parameters on energy structure, localization, and charge transport. We study quantities such as autocorrelation function, eigenspectra, density of states, Lyapunov exponents, transmission coefficients, and current-voltage curves. We show that the degree of sequence intricacy and the presence of correlations decisively affect the aforementioned physical properties. Periodic segments have enhanced transport properties. Specifically, in homogeneous sequences transport efficiency is maximum. There are several deterministic aperiodic sequences that can support significant currents, depending on the Fermi level of the leads. Random sequences is the less efficient category.

Figure 1

Classification of the DNA segments studied in this work based on the number and occurrence percentage of base-pair triplets. The boxes correspond to each of the eight possible triplets. For each segment, white boxes correspond to forbidden triplets, and the color of the rest corresponds to their occurrence percentage (calculated for large $N)$.

Figure 2

Scaling of the occurrence percentage of each TB parameter in various categories of DNA segments. (a) Periodic ${\left(\mathrm{GA}\right)}_m$. (b) ${\mathrm{TM}}_g$. (c) ${\mathrm{F}}_g$. (d) ${\mathrm{PD}}_g$. (e) ${\mathrm{RS}}_g$. (f) ${\mathrm{CS}}_g$. (g) ${\mathrm{GCS}}_g$. (h) ${\mathrm{KOL}}_g(1,2)$. (i) ${\mathrm{KOL}}_g(1,3)$. (j) Random $(50\%$ G, $50\%$ A).

Figure 3

Scaling of the autocorrelation function of various categories of DNA segments. (a) Periodic ${\left(\mathrm{GA}\right)}_m$. (b) ${\mathrm{TM}}_g$. (c) ${\mathrm{F}}_g$. (d) ${\mathrm{PD}}_g$ (e) ${\mathrm{RS}}_g$. (f) ${\mathrm{CS}}_g$. (j) ${\mathrm{GCS}}_g(4,2)$. (h) ${\mathrm{KOL}}_g(1,2)$. (i) ${\mathrm{KOL}}_g(1,3)$. (j) Random $(50\%$ G content, $50\%$ A content).

Figure 4

Eigenspectra and DOS of various periodic and quasiperiodic DNA sequences. (a) Periodic ${\left(\mathrm{GA}\right)}_m$. (b) ${\mathrm{TM}}_g$. (c) ${\mathrm{F}}_g$. (d) ${\mathrm{PD}}_g$. (e) ${\mathrm{RS}}_g$.

Figure 5

Eigenspectra and DOS of various fractal, Kolakoski and random DNA sequences. (a) ${\mathrm{CS}}_g$. (b) ${\mathrm{GCS}}_g(4,2)$. (c) ${\mathrm{KOL}}_g(1,2)$. (d) ${\mathrm{KOL}}_g(1,3)$. (e) Random $(50\%$ G and $50\%$ A).

Figure 6

Normalized IDOS of various categories of DNA segments. (a) Periodic ${\left(\mathrm{GA}\right)}_m$. (b) $\mathrm{TM}$. (c) $\mathrm{F}(\varphi$ is the golden ratio.) (d) $\mathrm{PD}$. (e) $\mathrm{RS}$. (f) $\mathrm{CS}$. (g) $\mathrm{GCS}$. (h) $\mathrm{KOL}(1,2)$. (i) $\mathrm{KOL}(1,3)$. (j) Random $(50\%$ G, $50\%$ A).

Figure 7

Lyapunov exponents of various categories of DNA segments. Top: Segments with $50\%$ G content: periodic ${\left(\mathrm{GA}\right)}_m$ (red dashed), $\mathrm{TM}$ [pink (light) filled], $\mathrm{KOL}(1,2)$ (blue dotted), and random [black (dark) filled]. Middle: Segments with more than $50\%$ G content: $\mathrm{F}$ [$61.82\%$, black (dark) filled], $\mathrm{RS}$ [$56.25\%$, pink (light) filled)], $\mathrm{PD}$ ($67.19\%$, red dashed), and random ($56.25\%$, blue dotted). Bottom: Segments with less than $50\%$ G content: $\mathrm{KOL}(1,3)$ [$40.00\%$, pink (light) dashed], $\mathrm{CS}$ [$13.17\%$, black (dark) filled], $\mathrm{GCS}(4,2)$ [$6.25\%$, red (dark) dashed], and two random [$40.00\%$ blue dotted, $10.00\%$ cyan (light) filled].

Figure 8

Transmission coefficients of various categories of DNA segments. Top: Segments with $50\%$ G content: periodic ${\left(\mathrm{GA}\right)}_m$ (red dashed), $\mathrm{TM}$ [pink (light) filled], $\mathrm{KOL}(1,2)$ (blue dotted) and random [black (dark) filled]. Middle: Segments with more than $50\%$ G content: $\mathrm{F}(61.82\%$, black/dark filled), $\mathrm{RS}\mathrm{[}56.25\%$, pink (light) filled], $\mathrm{PD}(67.19\%$, red dashed), and random $(56.25\%$, blue dotted). Bottom: Segments with less than $50\%$ G content: $\mathrm{KOL}(1,3)\mathrm{[}40.00\%$, pink (light) dotted], $\mathrm{CS}\mathrm{[}13.17\%$, black (dark) filled], $\mathrm{GCS}(4,2)(6.25\%$, red dashed), and two random $\mathrm{[}40.00\%$ blue (dark) dotted, $10.00\%$ cyan (light) filled].

Figure 9

The role of the leads' Fermi level, $E_M$, to the $I\text{−}V$ curve of a ${\left(\mathrm{GA}\right)}_{16}$ segment. The vertical dotted lines encompass the bands of the segment.

Figure 10

The $I\text{−}V$ curves of various categories of DNA segments for $E_M=-8.35$ eV. Categories as in Figs. 7 and 8. (a) Periodic ${\left(\mathrm{GA}\right)}_m,\mathrm{TM}$, and $\mathrm{KOL}(1,2)$ segments and a random segment with similar G content. (b) $\mathrm{F},\mathrm{PD},\mathrm{RS}$ segments, and a random segment with similar G content. (c) (top) $\mathrm{KOL}(1,3),\mathrm{CS},\mathrm{GCS}(4,2)$ segments; (bottom) random rearrangements of $\mathrm{KOL}(1,3),\mathrm{CS}$, and $\mathrm{GCS}(4,2)$ segments, respectively.

Figure 11

$I\text{−}V$ curves of various categories of DNA segments for $E_M=-7.95$ eV.

Figure 12

The $I\text{−}V$ curves of the ${\left(\mathrm{G}\right)}_{32}$ and ${\left(\mathrm{A}\right)}_{32}$ segments.