Theoretical Description of a DNA-Linked Nanoparticle Self-Assembly

Nanoparticles tethered with DNA strands are promising building blocks for bottom-up nanotechnology, and a theoretical understanding is important for future development. Here we build on approaches developed in polymer physics to provide theoretical descriptions for the equilibrium clustering and dynamics, as well as the self-assembly kinetics of DNA-linked nanoparticles. Striking agreement is observed between the theory and molecular modeling of DNA-tethered nanoparticles.

Figure 1

Temperature dependence of the fraction of bonded arms at equilibrium, $p_{\mathrm{eq}}$ for system sizes $N=200$ and $N=1000$. The solid line represents Eq. (1) with $\Delta U=1.78$ and $\Delta S=20.0$. The dotted line indicates percolation threshold predicted by FS theory. The inset shows a graphical representation of one threefold-functional particle bonding to one twofold-functional particle. Large red spheres indicate NPs, blue spheres indicate backbone beads of ssDNA, and small green spheres indicate sticky sites.

Figure 2

(a) Equilibrium distribution of finite-size clusters $N_k$. Points are simulation data (system size $N=1000$), and lines are the FS predictions. The dashed line represents the expected power-law decay $N_k\sim k^{-2.5}$ at percolation [31]. (b) Number of finite-size clusters $N_c$ as a function of $p$. Points are simulation data (system size $N=200$), and the black solid line is the FS prediction. The inset enlarges the region above percolation threshold (indicated by the dotted line).

Figure 3

(a) Diffusion constant $D$ as a function of inverse temperature. The dotted line indicates the location of the percolation threshold, and the red solid line represents Arrhenius behavior with an activation energy $\Delta U=1.78$—the same value reported for $p(T)$ in Fig. 1. The inset shows $D$ as a function of fraction of nonbonded arms. (b) Diffusion constant $D$ versus number of finite clusters $N_c$. The line shows the predicted behavior $D=D_1{N}_c/N$. The calculation of $⟨\Delta r^2⟩$ is averaged over 8–15 independent runs.

Figure 4

(a) Rate of bond formation $dp/dt$ as a function of fraction $p$ of bonded strands, and (b) time evolution of $p$ starting from a high-$T$ state [$T_{\mathrm{init}}=0.2$, $p(0)=0.008$]. The inset shows the $T$ dependence of the two rate constants. Symbols are simulation results (system size $N=1000$), averaged over 35 independent runs at short time and 15 runs at longer time. Dashed and solid lines represent, respectively, the prediction of Eq. (3) in the reaction-limit scenario (constant $k$) and with consideration of the $p{k}_0$ rate coefficient.