A stepwise assembly framework enables circuit topology optimization with tree search.

(A) Circuit topologies are built step-by-step by adding interactions until the game is ended by taking the “terminate” action (the STOP sign). (B) Each MCTS iteration undergoes four phases: (1) Selection: The UCT criterion is used to recursively select the most promising action from the current state si. (2) Expansion: If the edge has not been sampled yet, it is added to the tree. (3) Simulation: If the circuit has not been completed yet, take random assembly moves until “terminate” is chosen. Simulate the resulting topology with random parameters and evaluate the reward q based on a target phenotype. (4) Backpropagation: update the history of rewards rij over past visits υij for the visited edges . (C) The UCT policy for selecting balances exploitation and exploration based on past trials. The exploration term, tuned by a hyperparameter c, favors actions that are under-sampled relative to the alternatives. (D) Once assembled, a motif creates an enriched subgraph in the search graph.

Model rate parameters

CircuiTree efficiently identifies simple and robust 3-component oscillators.

(A) All three-component transcription factor (TF) circuits (3,325 up to symmetry) were enumerated with 104 random parameter sets (i) and evaluated for oscillations (ii) using an autocorrelation-based reward function. (B) A representative MCTS run. With more iterations (N), the search graph T (represented by a spanning tree for simplicity) expands to encounter more oscillators (orange circles) and improve its best predicted oscillator topology (shown in black). (C) A heatmap showing the average rate of discovery, or recall, for each oscillator (proportion of n = 50 replicates. Rows (oscillators) are sorted in order of complexity, or the number of interactions, and oscillators with the same complexity are sorted by descending robustness Q. Sparse oscillators are found before more complex ones, with a preference for the most robust candidates. (D) Precision (blue) and recall (orange) of oscillator classification (mean ± 95% CI, n = 50). CircuiTree’s recall is particularly high for the 10% most robust oscillators (red), reaching 94.7% after 105 iterations. See also Figures S1, S2, S3, and S4.

Dimensionless variables and limits imposed on random parameter sampling

Motifs identified from search results form a cluster of optimal 3-node oscillators.

A complexity atlas of oscillators with ≤3 components. Circles are oscillator topologies identified by enumeration, and edges link oscillators that differ by the addition/removal of one interaction. 97.7% of oscillators (216/221) are topologically related to one of the four motifs for 3-node oscillation, shown above the atlas in red boxes. Bold circle borders indicate oscillators found to be motifs based on enumeration. Circle color indicates the rate with which CircuiTree labels each oscillator as an assembly motif. Circle size indicates Qmotif, the average robustness for a circuit completed randomly starting from this state of the assembly game. The correlation between discovery rate and Qmotif (plotted in Figure S5B) suggests that motifs found by CircuiTree correspond to beneficial game states. The bolded edges, which connect oscillators with a discovery rate > 80%, form a contiguous cluster representing optimal assembly strategies. The most robust oscillator, the repressilator with PAR of all components, is shown on the bottom-left and indicated on the atlas by a green arrow. See also Figures S4 and S5.

Parallelized CircuiTree scales to large design spaces.

(A) A parallelized version of CircuiTree was used to search for five-node oscillators with ≤ 15 interactions (left) that oscillate despite a 50% chance of a single random deletion (right). (B) Search results after 5 · 106 iterations. Circles are putative oscillators , plotted on axes of robustness based on samples with or without a deletion . Error bars indicate standard error of the mean. Circle size and color indicate the number of samples υi and the overall robustness , respectively. Dashed lines show different theoretical values of fault-tolerance (FT), or the average number of deletions a circuit can sustain. Examples of circuits with different FT (labeled i-iii) are shown in (C). (D) Box plots of , grouped by the number of motifs in each putative oscillator. Multiple motifs, particularly Rep, increase robustness to deletions. (E) Simulated trajectories for the topologies in (C) for all single deletions. The 3AI+3Rep oscillator (FT ≈ 4/5) contains 6 interleaved oscillator motifs. See also Figures S6 and S7.

Motif multiplexing makes oscillators resistant to the failure of components.

The 3AI+3Rep circuit (A, top) oscillates with different limit cycles after partial knockdowns of different genes. (A, bottom) An exemplary trajectory of 3AI+3Rep is shown on axes of the first two principal components (PCs) of phase space. Transparent circles indicate the dominant species at each time-point. (B) Trajectories under scenarios where transcription rate is reduced by a factor KD. The ordering of species in the limit cycle at KD = 100% is shown by the inset diagram. (C) Oscillation quality and frequency in single-gene knockdowns. Oscillations persist (ACFmin < ACFthresh) for most knockdowns of genes B, C, D, and E (middle). Oscillation frequency (bottom) is pulled from its WT value in knockdowns of A, B, C, and D. (E) Robustness to parameter variation between TFs. The power spectral density of the trajectory of TF A (bottom, mean) and the overall oscillation rate (top, mean ± SEM) are shown for simulations in which parameters were perturbed by a Gaussian kernel of width σparam (n = 50 replicates). A dissipation of fundamental and harmonic frequencies and corresponding loss of oscillations occurs for σparam > 5 · 10−2.