For nanoscopic colloids, we suggest that such patches could be realized by building the particles through DNA origami. Such detailed potentials could be used to inform parameter choices for coarse-grained models of patchy particles 35 when the patches are made from patterned DNA coatings. Figure 5 shows the potentials of mean force of these plates as a function of plate separation and lateral displacement (zero lateral displacement means that the centers of the circular patches line up vertically). The only hybridization allowed is between strands on opposite plates, and we consider solution hybridization energies of −8 k B T, −7 k B T, and −6 k B T. As above, the strands are modeled as 20 nm rigid rods. Our fourth and final example in this section is a system of two parallel plates, each with a 100 nm-radius circular patch of dsDNA strands, distributed uniformly within the patch at a coverage density 1/(20 nm) 2. Importantly, since the full theory retains detailed information on the grafting points of c and c′ tethers, the MC results are again recovered to quantitative accuracy. The typical separation between grafting points that result from Poisson statistics is far larger than the length of the tethers, but these inhomogeneities are not captured by a mean field approach. The large discrepancies in the number of c– c′ bonds formed (black solid and dashed curves in Figure 4), and the resulting discrepancies in the potentials of mean force, result from the low coverage density of these tethers. The example also illustrates one way in which the mean-field approach fails, and how the full theory behaves for such cases. ![]() Our choices result in the interaction energy per unit area having up to two different minima, depending on the specific temperature (binding strength). Figure 4 summarizes the behavior of this system. Thus, our third system features three different types of rods, of different lengths, with different coverages and with a nontrivial interaction matrix. Finally, we set the grafting density of the c and c′ tethers to 1/(30 nm) 2. Furthermore, we set the lengths of the a, b, a′, and b′ tethers to 30 nm and those of the c and c′ tethers to 10 nm. The sticky ends of c and c′ hybridize only with each other and with the same strength as the a and a′ sticky ends. 16,25įor our third example, we generalize the setup of Figure 3 to include a third type of tether on each plate, respectively, c and c′. The supplementary material 26 presents an alternate derivation of the mean-field approximation to our theory using a saddle-point analysis, making explicit the connection between our theory and earlier successful but more specialized approaches. ![]() We show how a physically motivated and simple change to the equations used by these authors corrects their theory's principal shortcoming, and that the corrected version reduces to a specialization of our theory. Appendix B explains how our theory is connected to the local chemical equilibrium treatment recently introduced by Rogers and Crocker 19 (itself an improvement of the work of Biancaniello, Kim, and Crocker in Ref. 3 and 4, and 14)) and ideal chains (as used in Refs. We also summarize the results for short, stiff rods (which model short dsDNA tethers common in micron-sized DNACCs (Refs. ![]() In Appendix A, we discuss an important technical aspect of our theory, namely, how to calculate the entropic cost of binding for two general polymeric ligands.
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