Statistical mechanics of DNA-mediated colloidal aggregation

We present a statistical mechanical model of aggregation in colloidal systems with DNA-mediated interactions. We obtain a general result for the two-particle binding energy in terms of the hybridization free energy $\mathrm{\Delta }G$ of DNA and two model-dependent properties: the average number of available DNA bridges $⟨N⟩$ and the effective DNA concentration $c_{\mathrm{eff}}$. We calculate these parameters for a particular DNA bridging scheme. The fraction of all the $n$-mers, including the infinite aggregate, are shown to be universal functions of a single parameter directly related to the two-particle binding energy. We explicitly take into account the partial ergodicity of the problem resulting from the slow DNA binding-unbinding dynamics, and introduce the concept of angular localization of DNA linkers. In this way, we obtain a direct link between DNA thermodynamics and the global aggregation and melting properties in DNA-colloidal systems. The results of the theory are shown to be in quantitative agreement with two recent experiments with particles of micron and nanometer size.

Figure 1

(Color online) Graphical depiction of various schemes for DNA bridging. (A) A freely jointed, rigid bridge constructed from complementary linker DNA. (B) A flexible bridge can be constructed using complementary linker DNA. (C) A rigid bridge constructed from short, flexible linker DNA and a long, rigid linker.

Figure 2

(Color online) The statistical weight of a bound state is calculated by determining the number of hybridized configurations for two complementary linker chains relative to the number of unhybridized configurations.

Figure 3

(Color online) Cross-sectional view of the hybridization of two complementary rigid linker DNA. The effective concentration is calculated in a planar approximation to the particle surface.

Figure 4

(Color online) The actual unbound fraction $f$ is the concatenation of the aggregate profile for $T<T^{*}$and the $n$-mer profile for $T>T^{*}$. The fraction of particles in dimers, trimers, and tetramers is also plotted.

Figure 5

(Color online) Comparison of the melting curves $f\left(T\right)$ determined by our model to the experimental data of Chaikin et al. (see Fig. 2 in Ref. 9). The four data sets are for the four different polymer brushes used. For the model fits we find that $⟨N{⟩}_{2+}=2.01$ for crosses, 2.07 for solid triangles, 2.13 for empty triangles, and 2.35 for squares.

Figure 6

(Color online) The effect of the linker DNA grafting density $\sigma$ on the melting profile $f\left(T\right)$. The results of the model are compared with experimental data in Ref. 12. The three data sets represent grafting densities of 100% (squares), 50% (circles), and 33% (triangles) for which $⟨N{⟩}_{2+}=2.32$, 2.16, and 2.05, respectively.