Simultaneous optimization of assembly time and yield in programmable self-assembly

TL;DR

The paper addresses the challenge of predicting and controlling self-assembly kinetics in programmable systems, with a focus on the semiaddressable regime where nondeterministic binding creates off-target kinetic traps. It introduces a framework that treats assembly as a complex reaction network, deriving rate expressions from bond energies and diffusion and then optimizing both the bond energies $E$ and particle concentrations to improve kinetics without sacrificing equilibrium yield. The results show substantial speedups—often by orders of magnitude—across a range of target structures, with the largest gains in nondeterministic, highly interconnected designs and notable improvements in avoiding kinetic traps. The work demonstrates a practical, generalizable method for simultaneous optimization of kinetics and yield in programmable self-assembly, highlighting the value of semiaddressability for faster, cheaper, and more reliable assembly across nanotechnologies and biomolecular systems.

Abstract

Rational design strategies for self-assembly require a detailed understanding of both the equilibrium state and the assembly kinetics. While the former is starting to be well understood, the latter remains a major theoretical challenge, especially in programmable systems and the so-called semiaddressable regime, where binding is often nondeterministic and the formation of off-target structures negatively influences the assembly. Here, we show that it is possible to simultaneously sculpt the assembly outcome and the assembly kinetics through the underexplored design space of binding energies and particle concentrations. By formulating the assembly process as a complex reaction network, we calculate and optimize the tradeoff between assembly speed and quality, and show that parameter optimization can speed up assembly by many orders of magnitude without lowering the yield of the target structure. Although the exact speedup varies from design to design, we find the largest speedups for nondeterministic systems where unoptimized assembly is the slowest, sometimes even making them assemble faster than optimized fully-addressable designs. Therefore, these results not only solve a key challenge in semiaddressable self-assembly, but further emphasize the utility of semiaddressability, where designs have the potential to be faster as well as cheaper (fewer particle species) and better (higher yield). More broadly, our results highlight the importance of parameter optimization in programmable self-assembly, and provide practical tools for simultaneous optimization of kinetics and yield in a wide range of systems.

Figures (3)

• Figure 1: Assembly kinetics as a complex reaction network. (a) Binding rules showing selectivity of binding. Only particle sides that are connected by edges may bind to each other, thereby limiting what structures may form. (b) From the binding rules, we enumerate all possible structures, as illustrated here by sketching all possible monomers, dimers, trimers, and so on. For the rules shown in (a), only 71 structures are possible, the largest of which is shown on the right. (c) To investigate the assembly kinetics, we construct all possible reactions between all structures. Shown here is one of 268 possible reaction pairs resulting from the binding rules in (a). The rate constant for two structures $a$ and $b$ to combine into a structure $c$ is given by $k_{ab\to c}$, whereas the fragmentation of $c$ into $a$ and $b$ is controlled by the rate $f_{c \to ab}$. (d) Time evolution of the assembly process for uniform binding energies fixed at $12 \, k_\mathrm{B}T$ (dashed lines), and optimized unequal binding energies (solid lines), as discussed in the text. Shown are the time dependent yields of three monomers (blue, red, yellow), an off-target four-particle square (purple), and the seven-particle target structure (teal). Total particle concentration is $0.01/\sigma^3$. Time is measured in units of the monomer diffusion time, $\tau_\mathrm{D}$.
• Figure 2: Yield and equilibration time tradeoff. Equilibrium yield as a function of equilibration time, for the seven-particle square shape (a), a 14-particle triangle shape (b), and a ten-particle hexagon shape (c). The different curves correspond to different methods of parameter optimization. Black solid curve: unoptimized, uniform binding energies and stoichiometric particle concentrations. Red solid curve: optimized parameters obtained by minimizing the correlation time $\tau_\mathrm{c}$ through the procedure described in Section \ref{['sec:minimize']}, while keeping total particle concentration fixed at $0.01 / \sigma^3$. (d) Relationship between equilibration time (here defined to be the time $\tau_{97}$ where target yield first reaches $97\%$ of the equilibrium value) and correlation time $\tau_\mathrm{c}$ for different methods of parameter optimization (colors). Marker shape corresponds to the system: squares correspond to (a), triangles correspond to (b), and hexagons to (c). The black line has slope 1 and is shown for comparison.
• Figure 3: Addressability and equilibration time. (a) Favorable example showing the equilibration time $\tau_{97}$ as a function of the number of distinct particle species for different designs of the same target shape. At all data points, the final yield of the target structure is 90% and the total particle concentration is $0.1 / \sigma^3$. Unoptimized parameters (black curve) have uniform binding energies across all programmable bonds and stoichiometric concentrations, optimized parameters (red curve) are found using the approach described in Section \ref{['sec:minimize']}. (b) The number of off-target structures for different designs of the same target shape. Here, off-target structures are defined as any possible arrangement of particles that is not a substructure of the target. Empty symbols indicate deterministic designs, where the number of off-targets is zero. (c) Histogram of the assembly speedup ratio for 2932 deterministic systems. (d) Histogram of the assembly speedup ratio for 1137 nondeterministic systems. (e) Scatter plot showing the relationship between speedup ratio and the initial, unoptimized equilibration time. Each marker corresponds to one design. Marker color indicates the size (number of particles) in the target structure. Diamond markers correspond to fully addressable designs, square markers to semiaddressable and deterministic designs, circular markers to semiaddressable and nondeterministic designs. Empty markers correspond to systems where the target structure is a tree. Using optimized interactions, systems below the red dashed line equilibrate in less than one day using particles of $20 \, \mathrm{nm}$ radius (see main text).