Extended Data Fig. 4: Stochastic Greedy Model of nucleation, based on repeated stochastic simulations.

a, The frequently-used kinetic Tile Assembly Model (kTAM)^59,60 has rates for tile attachment and detachment events based on tile and assembly diffusion and total binding strength of correct attachments a tile can make at a lattice site. Here u₀ = 1 M. b, These rates can be used to derive a free energy for any tile assembly in a system, and, assuming fixed monomer concentrations, an equilibrium concentration for any assembly. Schulman & Winfree³⁷ showed that the equilibrium concentration of the highest-energy assembly along a nucleation trajectory under this assumption provides an upper bound for nucleation rate through that trajectory, with or without fixed monomer concentrations. However, in a large system, considering all possible intermediate assemblies and all pathways, including many that are extremely unlikely, would be infeasible. Thus, we developed the Stochastic Greedy Model (SGM) to generate stochastically-chosen paths of tile attachments. c, Starting from a single tile (chosen with probability proportional to relative concentration), whenever the assembly is in a state A_stable where there is no tile attachment that would be favorable (have ΔG < 0), one of the possible unfavorable (with ΔG ≥ 0) attachments is stochastically chosen, resulting in a higher-G state A_unstable. Then, all subsequent possible ΔG < 0 attachments are made, resulting in the next \(A{{\prime} }_{{\rm{stable}}}\) state; for our system of unique tiles for each site in the lattice, this sequence of favorable steps has a unique resulting assembly. d, The process repeats until all tiles in a shape are attached, which results in a trajectory with a maximum-G assembly that can be used to bound the rate of nucleation, η, through that particular trajectory. e, By using this process to collect many trajectories, and then repeating the entire process for each of the three shapes in the system, we can estimate nucleation rates dependent upon temperature, with the assumption that tile monomer concentrations do not deplete, and that the trajectories found are a reasonable representation of likely trajectories. For comparison between model predictions and experimental data in Extended Data Figs. 6d and 9b, we determined the temperature at which the model predicted the nucleation rate exceeded a threshold (orange line), to compare with when fluorescence quenching exceeded a threshold. For details on the SGM model, see Supplementary Information section 2.2. f, To study the winner-take-all effect, we use a simplified chemical reaction network (CRN) model for the case of systems with shared tiles (shown here) and a similar model for systems without shared tiles (described in Supplementary Information section 2.3). Here, \({c}_{n}^{H}\) represent tiles in the flag area of shape H, which have initially higher concentrations; \({c}_{n}^{A}\) are the corresponding tiles in the flag area of shape A, which have normal concentrations; and c_g represent tiles involved in growth from the nucleated seed H_nuc to the almost-complete structure H_mid; and similarly for structure A. A more detailed model based on (but simpler than) the SGM gives qualitatively similar results, as detailed in Supplementary Information section 2.3.