Fig 1.
Assembly sequence from genotype to phenotype in the standard polyomino self-assembly model.
The full sequence of generating a phenotype from a genotype for deterministic (left) and nondeterministic (right) assemblies. The binding sites on the subunits are transcribed from the genotype in a clockwise fashion. The assembly graph encodes all possible interactions (0s noninteracting, 1s and 2s interact with each other, 3s and 4s interact with each other, etc.) among the subunits, indicated by solid lines. In the case of nondeterministic genotypes, different polyominoes may emerge as the outcomes of the stochastic assembly process. Here we perform 10 repeated assemblies, and define the phenotype of a genotype as the polyomino that appears most often. Other definitions of a phenotype from the distribution of polyominoes are also possible.
Fig 2.
(a) Explicit subsite interactions (dotted lines) between two binding sites, showing the “head to tail” alignment. The Hamming distance between the counter-aligned sites is 4, and so the interaction strength is . (b) Taking the critical strength
, these two subunits encode two interactions in the assembly graph. The interactions have different strengths (indicated by line thickness), with the upper interaction stronger (
) than the lower (
).
Fig 3.
Example system of six assembly graphs.
The interactionless initial condition and an example system of six assembly graphs with associated polyominoes. The assembly graphs (and polyominoes) are grouped into vertical columns that are ordered by the number of interactions (from left to right: one, two, and three interactions). Three assemblies are nondeterministic, and are marked with a *. In the nondeterministic cases we only show the most common polyomino structure, which also corresponds to our formal definition of the phenotype.
Fig 4.
Each box corresponds to a different phenotype, with marker styles indicating interaction topology. Line colours (online) match the box colour of the direct ancestor, with “open” markers (print) indicating the ancestor is from an upper panel. Because phenotypes can transition at markedly different times (with 4% of simulations never discovering the rightmost phenotypes), the dynamics were aligned by counting the generations after that phenotype was discovered. Individual simulations are noisy due to their stochastic nature, but averaging over 10,000 simulations yielded the stable trends shown here. Values for within individual simulations typically fluctuated within ±0.01 of the mean. Mean trends stabilized quickly, so results were truncated after 250 generations. Black dashed lines in the panels are from the Markov prediction. The * again indicates the three nondeterministic assemblies. Interface strengths in deterministic assemblies evolve predictably, while nondeterministic assemblies diverge rapidly.
Fig 5.
Decision tree for heterotetramer.
The assembly graph has interaction strengths A and B. Each seed is a starting point for the decision tree, incrementally progressing until assembly terminates. In this situation, once a gray subunit is placed, assembly deterministically ends with the heterotetramer, rendering further branching unnecessary. The lower branchings have an extra weighting factor of two, due to two indistinguishable assembly steps.
Fig 6.
Phenotype transition success and ancestry.
(a) Transitions to deterministic assemblies have high success, tending to perfect in an infinite population. Conversely, transitions to nondeterministic assemblies (marked with *) typically have less success. Transition rates between nondeterministic assemblies vary considerably, due to the varying overlap between the interfaces of an ancestor and the stronger interfaces of the descendant. Interaction strength is indicated by line thickness. The transition locations in phase space of ancestors are shown for the heterotetramer and 12-mer in (b) and (c) respectively. (b) For transitions from both the dimer and homotetramer, one bond has been strengthened through evolution (black) and one is new and at the critical interaction strength (gray). Compared to the evolutionary equilibrium of the heterotetramer, the dimer has a much more favorable ratio of strengths than the homotetramer, as indicated by its closer position in phase space. Likewise in (c), the evolutionary equilibrium of the octomer has much more similar ratios of interaction strength to the 12-mer than the heterotetramer has. In addition to the heterotetramer being further down the determinism gradient, it more frequently misassembles the phenotype, lowering its transition success even further.
Fig 7.
Generalized interactions are not always transitive.
In the generalized model, knowledge of one interaction does not fix the binding sites of another related interaction. Earlier in the nondeterministic case in Fig 1, this assembly graph had A = 1, B = 2 fixing ? = 1. Here, choosing binding sites A and B still leaves 5 possibilities for ‘?’, taking Sc = .75. The possibilities marked with † self-interact, and so would technically add an interaction to the assembly graph.
Fig 8.
Interaction strengths can adapt to changing fitness landscapes.
Periodically alternating the fitness landscape produces cyclic behaviour in interface strengths. Despite starting from a range of initial conditions, all simulations eventually converge to the optimal path to transition between the 10-mer and 12-mer and back. The change in fitness landscape is indicated by the red or blue colours, with arrows indicating the direction of flow. Both phenotypes are produced with the same three interactions; it is only the relative ordering of interaction strength that matters. A breakdown of each fitness landscape and local gradients can be seen in S2 Fig.