Self-assembly of a dimer system

Programming interactions in magnetic handshake materials

The ability to rapidly manufacture building blocks with specific binding interactions is a key aspect of programmable assembly. Recent developments in DNA nanotechnology and colloidal particle synthesis have significantly advanced our ability to create particle sets with programmable interactions, based on DNA or shape complementarity. The increasing miniaturization underlying magnetic storage offers a new path for engineering programmable components for self assembly, by printing magnetic dipole patterns on substrates using nanotechnology. How to efficiently design dipole patterns for programmable assembly remains an open question as the design space is combinatorially large. Here, we present design rules for programming these magnetic interactions. By optimizing the structure of the dipole pattern, we demonstrate that the number of independent building blocks scales super linearly with the number of printed domains. We test these design rules using computational simulations of self assembled blocks, and experimental realizations of the blocks at the mm scale, demonstrating that the designed blocks give high yield assembly. In addition, our design rules indicate that with current printing technology, micron sized magnetic panels could easily achieve hundreds of different building blocks.

Derangement model of ligand-receptor binding

We introduce a derangement model of ligand-receptor binding that allows us to quantitatively frame the question “How can ligands seek out and bind to their optimal receptor sites in a sea of other competing ligands and suboptimal receptor sites?” To answer the question, we first derive a formula to count the number of partial generalized derangements in a list, thus extending the derangement result of Gillis and Even. We then compute the general partition function for the ligand-receptor system and derive the equilibrium expressions for the average number of bound ligands and the average number of optimally bound ligands. A visual model of squares assembling onto a grid allows us to easily identify fully optimal bound states. Equilibrium simulations of the system reveal its extremes to be one of two types, qualitatively distinguished by whether optimal ligand-receptor binding is the dominant form of binding at all temperatures and quantitatively distinguished by the relative values of two critical temperatures. One of those system types (termed “search-limited,” as it was in previous work) does not exhibit kinetic traps and we thus infer that biomolecular systems where optimal ligand-receptor binding is functionally important are likely to be search-limited.