Translocation of polymers into crowded media with dynamic attractive nanoparticles

The translocation of polymers through a small pore into crowded media with dynamic attractive nanoparticles is simulated. Results show that the nanoparticles at the trans side can affect the translocation by influencing the free-energy landscape and the diffusion of polymers. Thus the translocation time $\tau$ is dependent on the polymer-nanoparticle attraction strength $\varepsilon$ and the mobility of nanoparticles $V$. We observe a power-law relation of $\tau$ with $V$, but the exponent is dependent on $\varepsilon$ and nanoparticle concentration. In addition, we find that the effect of attractive dynamic nanoparticles on the dynamics of polymers is dependent on the time scale. At a short time scale, subnormal diffusion is observed at strong attraction and the diffusion is slowed down by the dynamic nanoparticles. However, the diffusion of polymers is normal at a long time scale and the diffusion constant increases with the increase in $V$.

Figure 1

A two-dimensional side view of the schematic representation of polymer translocation through a nanopore in the membrane at $x=0$. Attractive NPs distributed at the trans side move randomly with mobility $V$. There is no NP at the cis side.

Figure 2

The mean translocation time $\langle \tau \rangle$ as a function of polymer-NP interaction $\varepsilon$ for different parameters: polymer length $N$ (a), chemical potential difference $\mathrm{\Delta }\mu$ (b), NP concentration ${\varphi }_{\mathrm{t}}$ (c), and NP mobility $V$ (d). Solid lines in (a) are parallel to the $x$ axis.

Figure 3

Log-log plot of the mean translocation time $\langle \tau \rangle$ as a function of the NP mobility $V$ for polymers at different NP concentrations ${\varphi }_{\mathrm{t}}$ and at $\varepsilon =1.5$ (a) and for polymers at different polymer-NP attractions $\varepsilon$ and at ${\varphi }_{\mathrm{t}}=0.01$ (b). Other parameters are polymer length $N=100$ and chemical potential difference $\mathrm{\Delta }\mu =-0.5$. Solid lines are guides for the eyes.

Figure 4

The distribution probability of contact time $P\left({\tau }_{\mathrm{c}}\right)$ for weak polymer-NP attraction $\varepsilon =0.6$ and strong attraction $\varepsilon =1.8$. Insets: the dependence of mean contact time $\langle {\tau }_{\mathrm{c}}\rangle$ on the polymer-NP interaction at $V=0.1$ (a) and on the NP mobility at $\varepsilon =1.5$ (b). Parameters: polymer length $N=100$, chemical potential difference $\mathrm{\Delta }\mu =-0.5$, and NP concentration at the trans side ${\varphi }_{\mathrm{t}}=0.01$.

Figure 5

(a) Mean number of polymer-contacted NPs $\langle N_{\mathrm{c}}/N\rangle$ vs the mobility $V$ at attractive strength $\varepsilon =1.5$. Inset: the number of polymer-contacted NPs $N_{\mathrm{c}}$ varies with time $t/\tau$ during the translocation process at different NP mobilities $V$ = 0, 0.02, and 1.0 (from bottom to top). (b) Variation of the mean number of polymer-NP pairs $\langle N_{\mathrm{p}}\rangle$ and the mean number of polymer-contacted NPs $\langle N_{\mathrm{c}}\rangle$ at the end of the polymer translocation with the interaction $\varepsilon$. Other parameters: polymer length $N=100$, chemical potential difference $\mathrm{\Delta }\mu =-0.5$, and NP concentration ${\varphi }_{\mathrm{t}}=0.01$.

Figure 6

The dependence of the mean translocation time $\langle \tau \rangle$ on average contact energy $\langle N_P\rangle \varepsilon$ for different NP mobilities $V$ = 0.05, 0.1, 0.5, and 1.0 at $\mathrm{\Delta }\mu =0$ and $-0.5$. Here polymer length $N=100$ and NP concentration ${\varphi }_{\mathrm{t}}=0.01$.

Figure 7

The dependence of the mean-square end-to-end distance $\langle R^2\rangle$ (black open circles) and the mean-square radius of gyration $\langle R_G^2\rangle$ (red solid circles) on the polymer-NP interaction $\varepsilon$ at $V=1$ (a) and on the NP mobility $V$ at $\varepsilon =1.5$ (b). Polymer length is $N=100$, chemical potential difference $\mathrm{\Delta }\mu =-0.5$, and NP concentration ${\varphi }_t=0.01$. Solid lines are guides for the eyes.

Figure 8

(a) The dependence of asphericity parameter $\langle A\rangle$ on relaxation time $t$ at different NP mobilities: $V$ = 0 (thin solid black line), 0.2 (thick solid red line), and 0.8 (dashed blue line). Here the polymer-NP interaction is $\varepsilon =1.5$. (b) Evolution of $\langle A\rangle$ at different polymer-NP interactions: $\varepsilon$ = 0 (thin solid black line), 1.0 (thick solid red line), 1.3 (dashed blue line), and 1.8 (dotted purple line). Here the NP mobility is $V=1$. Other parameters: polymer length $N=100$ and NP concentration $\varphi =0.01$.

Figure 9

The dependence of mean asphericity parameter $\langle A\rangle$ (a) and mean number of polymer-NP pairs $\langle N_{\mathrm{p}}\rangle$ (solid circle) and mean number of polymer-contacted NPs $\langle N_{\mathrm{c}}\rangle$ (open square) (b) on polymer-NP interaction $\varepsilon$ for polymers in a crowded environment. Parameters: polymer length $N=100$, NP concentration $\varphi =0.01$, and NP mobility $V=1$.

Figure 10

Log-log plot of the mean-square displacement of the center of mass $\langle \mathrm{\Delta }{r}^2\rangle$ vs simulation time $t$ at different NP concentrations $\varphi$ and polymer-NP interactions $\varepsilon$. Other parameters: polymer length $N=100$ and NP mobility $V=0.1$. The straight lines with slope 1.0 indicate the normal diffusion of polymers.

Figure 11

Log-log plot of the mean-square displacement of the center of mass $\langle \mathrm{\Delta }{r}^2\rangle$ vs simulation time $t$ at different NP mobilities $V$. Other parameters: polymer length $N=100$ and polymer-NP interactions $\varepsilon =1.5$. The straight line with slope 1.0 indicates the normal diffusion of polymers.