Filling of the one-dimensional lattice by $k$-mers proceeds via fast power-law-like kinetics

During the filling of a one-dimensional lattice by $k$-site-long hard particles, we find that after initial “jamming” a power-law-like decay of the density of interparticle gaps occurs, described by a much larger exponent than that expected from mean-field theory. We show that this effect dominates post-jamming filling for large $k$ and should be observable, e.g., during the binding of proteins along a long, stretched DNA molecule.

Figure 1

(Color online) (a) Shown are adsorption(on), desorption(off), and diffusion events. (b) The gap density versus time for $k=20$; $r_{\text{off}}∕r_{\text{on}}={10}^{-5}$, $r_{\mathrm{d}}=0$, and $L=20\times {10}^5$ lattice sites. Four regimes are observed: a rapid initial filling, jamming (constant $y$), an ordering regime with power-law-like decay of $y$, and an exponential decay to the equilibrium. The straight line with a slope $\approx 0.175$ is a guide to the eye, showing that the decay is much faster than the expected late time exponent $z=1∕19$.

Figure 2

(Color online) The gap density decay in adsorption-desorption mediated filling of 10 mers on a lattice for $r_{\text{off}}∕r_{\text{on}}$ values (a, green) ${10}^{–4}$, (b, blue) ${10}^{–5}$, (c, pink) ${10}^{–7}$, and (d, red) ${10}^{–10}$, with an apparent exponent 0.2. The power-law-like decay persists up to a density $\rho \approx 95\%$. A line with a $\text{slope}=–1∕9$, corresponding to the expected late time mean-field exponent, is plotted to demonstrate the far faster decay of the gap density. Note that even at a filling of $\approx 98\%$, the decay is faster than $–1∕9$. The arrows at the right indicate the corresponding equilibrium densities. The straight line (black) with a slope $=–0.2$ is a guide to the eye.

Figure 3

(Color online) The adsorption and desorption of $k$ mers on a lattice: the apparent power-law exponents for gap density decay, observed soon after jamming, as a function of $k$ (filled squares). As $k\to \infty$ the exponent goes to 0.15. The line passing through the filled squares is a guide to the eye. Note that the adsorption-desorption exponents are above the mean-field adsorption-diffusion result $z\left(k\right)=1∕(k-1)$, for all $k>4$. Inset shows the power-law regime in the $y\left(t\right)$ for $k=2$ (red), 4 (green), and 10 (blue).

Figure 4

(Color online) The gap density versus time for the continuum model; $r_{\text{off}}∕r_{\text{on}}={10}^{-7}$, $r_{\mathrm{d}}=0$, and $L={10}^5$. Three regimes are observed: a rapid initial filling, jamming (constant $y$), and an ordering regime with power-law-like decay of $y$. Inset: an expanded plot of apparent power-law regime; the linear fit gives the exponent $z=0.151\pm \mathrm{0.000\hspace{0.17em}18}$.

Figure 5

(Color online) The gap density versus time for $r_{\text{off}}∕r_{\text{on}}=$ values (a, red) ${10}^{-13}$, (b, green) ${10}^{-7}$, (c, pink) ${10}^{-5}$, and (d, blue) ${10}^{-4}$. The straight line with $\text{slope}=-0.15$ is given to compare the decay soon after jamming. At late times (below $y\approx 0.1$), the decay follows the mean-field curve as shown in Ref. [12]. Inset: The gap density decay for $r_{\text{off}}∕r_{\text{on}}={10}^{-13}$ is compared with the mean-field expression [curve e, $y=1∕\ln \mathbf{\left(}\alpha t\ln \left(\alpha t\right)\mathbf{\right)}$]. $\alpha$ is chosen in such a way that the slope at the beginning matches. The straight line is a fit to the first power-law regime $(\text{slope}=-0.15)$.

Figure 6

(Color online) The gap density dynamics of 10-mer. Numerical solution of Eq. (3) (a, green open circles) and the mean-field equation from Ref [7] (blue filled squares) are plotted for $\mathrm{k}=10$. This is compared with the Monte Carlo simulations result (b, pink). In the regime shown, all these results show a decay which is faster than the expected late time power-law decay $y\sim t^{-1∕9}$ (thin red straight line). The thicker black straight line has a slope $=-0.2$. Curves have been shifted to line up vertically.