Chain dynamics and power-law distance fluctuations of single-molecule systems

Chain-dynamics-induced distance fluctuations between any two points in a finite chain with or without cross links are investigated. This model leads to three regimes of temporal behavior for distance autocorrelation: (i) initial flat time dependence, (ii) $t^{-\alpha }$ power law, and (iii) long-time exponential decay. For an ideal Rouse chain with frequency-independent friction, $\alpha =\frac{1}{2}$. The span of the characteristic power-law behavior of a long chain could be reduced significantly with the presence of cross links.

Figure 1

Schematic diagram of a Rouse model with a chain of beads coupled by a spring to their nearest neighbors, containing a donor $\left(D\right)$ and an acceptor $\left(A\right)$.

Figure 2

(a) $C_Q\left(t\right)∕C_Q\left(0\right)$ of Eq. (4) for a $D$-$A$ pair attached to various positions $(n_D,n_A)$. (b) Semilogarithmic plot of $C_Q\left(t\right)∕C_Q\left(0\right)$ showing long-time exponential decay which is dictated by $\mid n_D-n_A\mid$. The decay rate is insensitive to the attached position, and $\mid n_D-n_A\mid =200$ is used. $1∕\Gamma$ was set at 1 for all illustrations as a time unit.

Figure 3

(a) $C_Q\left(t\right)∕C_Q\left(0\right)$ of Eq. (6) for various bead units $N$. $C_Q\left(t\right)∕C_Q\left(0\right)$ stays at 1 if $\Gamma t⪡1$, becomes a power law of $t^{-1∕2}$ when $\Gamma t⪢1$, and then breaks down much later. (b) Semilogarithmic plot of $C_Q\left(t\right)∕C_Q\left(0\right)$ showing a single-exponential decay at long times as $\exp (-t∕T_D)$ where $T_D=4N^2∕\Gamma {\pi }^2$.

Figure 4

Comparison of $C_Q\left(t\right)∕C_Q\left(0\right)$ among the Rouse model with a distributed $\Gamma$ (curve $A$, with $\Gamma$ set at $1∕\pi {\tau }_c$), the ordinary Rouse mode (curve $B$), and the FBN model (curve $C$) with $\exp (\Gamma t∕4)\mathrm{erfc}\left(\sqrt{\Gamma t∕4}\right)$.

Figure 5

(a) $C_Q\left(t\right)∕C_Q\left(0\right)$ for the Rouse model with a frequency-dependent friction and $\overline{\Gamma }\left(s\right)={\Gamma }_0(s+{\Gamma }_1{)}^{-\eta }{s}^{\eta }$, following an asymptotic power law of $t^{-\alpha }$, where $\alpha =(1-\eta )∕2$. For frequency-independent friction, $\eta =0$ and $\alpha =\frac{1}{2}$. Here ${\Gamma }_0={\Gamma }_1=1$ was used. (b) $C_Q\left(t\right)∕C_Q\left(0\right)$ for a chain of 250 units, ensemble averaged over random positions for various percentages of cross-linked beads. In addition to the nearest-neighbor couplings, a 10% of cross links, as an example, means that 25 pairs of randomly chosen beads are coupled by the harmonic spring constant.