1. Weak coupling (sub-threshold).
When the coupling parameters and are small, the RNA and protein populations fail to stabilize each other. Time-series simulations show both and declining toward near-extinction levels. This regime corresponds to unsynchronized dynamics, where mutual dependence is insufficient to sustain coadaptation.
2. Intermediate coupling (oscillatory regime).
At moderate values of coupling strength, the system does not converge to a fixed point but instead exhibits sustained oscillations. RNA and protein abundances rise and fall in tandem, locked in a perpetual cycle. Phase-plane trajectories reveal closed orbits characteristic of a limit cycle attractor, consistent with a Hopf bifurcation. Biologically, this corresponds to a molecular Red Queen regime, in which continual adaptation of RNA and proteins is required to maintain functional compatibility.
3. Strong coupling (stable attractor).
At higher coupling strengths, the system converges to a stable interior equilibrium. Both RNA and protein populations reach finite, mutually reinforcing steady states. This coadapted attractor is resilient to perturbations, echoing the stability of ribonucleoprotein complexes such as ribosomes. The simulations confirm the analytical prediction that coupling strength serves as a critical control parameter governing system dynamics. As coupling increases, the system transitions from collapse to oscillatory coevolution and finally to stable coadaptation. These transitions exemplify the bifurcation structure of molecular coevolution, where feedback and trade-offs generate emergent attractors with distinct evolutionary implications.
C. Empirical alignment with proteomic and genomic data
The simulation results align with several empirical signatures documented in comparative genomics and proteomics. This correspondence supports the plausibility of the CAS framework as a mechanistic explanation for RNA--protein coevolution.
1. Dipeptide frequency correlations.
The oscillatory regime predicted by the model implies fluctuating covariation between codon assignments and amino acid pairings. Empirical studies of dipeptide frequencies across diverse proteomes (as in the Journal of Molecular Biology dipeptide analysis) reveal persistent, non-random correlations between codon usage and structural motifs. These can be interpreted as molecular "fossils" of Red Queen--like cycles in which codon--protein pairings were periodically reshaped under coevolutionary dynamics.
2. Thermostability patterns.
The strong-coupling attractor regime corresponds to stable RNA--protein coadaptation optimized for robustness. Observed biases in thermophilic organisms, where codon usage is strongly correlated with thermostable dipeptides, match the model's prediction that attractors are shaped by environmental parameters such as temperature.
3. Genomic signatures of feedback.
Comparative genomics reveals conserved RNA motifs that consistently associate with specific protein domains, such as rRNA--protein complexes in ribosomes. These associations correspond directly to the model's bipartite mapping of RNA motifs to protein domains, reinforcing the idea of attractor-based modularity.
4. Evidence of punctuated transitions.
Molecular phylogenies often show abrupt shifts in codon reassignment and amino acid usage. These discontinuities are consistent with saddle-node bifurcations in the CAS model, where small parameter changes (e.g., mutation rate, resource availability) cause the system to jump between distinct coadapted states.
5. Red Queen dynamics at the molecular scale.
The existence of limit cycles in simulations parallels observed coevolutionary patterns, such as continual adjustments in tRNA modification and aminoacyl-tRNA synthetases. These empirical phenomena exemplify a molecular-scale Red Queen dynamic in which RNA and proteins co-adapt without reaching a static endpoint. Taken together, the CAS model captures not only abstract dynamical possibilities but also empirical regularities in real molecular data. By reproducing observed correlations and discontinuities, the model provides a mechanistic foundation for patterns that descriptive or statistical approaches have thus far only catalogued.
VI. Discussion
