Phenotypic complexity determines the predictability of molecular convergence

Whether the outcomes of evolution are predictable remains a central question in evolutionary biology. Convergent evolution, in which similar phenotypes arise independently in distinct lineages, is often interpreted as evidence of predictability under shared selective pressures. However, empirical studies have shown that convergent phenotypes can exhibit similar or disparate molecular bases, and no general framework predicts when convergence should occur. Here, we show that phenotypic complexity determines the extent of convergence under shared conditions. Using a deliberately minimal abstraction linking genotype and phenotype and evolutionary simulations, we investigated how regulatory gene expression programs evolve under selection for shared phenotypic optima. We find that phenotypes of low complexity can be produced by diverse regulator expression programs, permitting phenotypic convergence despite molecular divergence. However, as the complexity of the target phenotype increases, the variety of viable regulatory solutions decreases, funneling independently evolving lineages towards similar expression programs. In this regime, molecular convergence emerges as a predictable consequence of selection acting on finite regulatory repertoires, consistent with the phenomenon of "deep homology" observed in structures like eyes and limbs. Allowing evolutionary changes in regulator-target interactions relaxes these constraints and enhances divergence in expression between phenotypically convergent species. Together, our results provide general expectations for predicting when phenotypic convergence should be accompanied by convergent genetic mechanisms, and shed light on the causes of convergence in a broad range of phenotypic traits.