xix_ixi: frequency of type iii,
fjf_jfj: fitness of type jjj,
QjiQ_{ji}Qji: mutation probability from type jjj to type iii,
=kxkfk\phi = \sum_{k} x_k f_k=kxkfk: average fitness.
2. Coupled RNA--protein dynamics.
We define two coupled populations:
xiRx_i^RxiR: frequency of RNA sequence type iii,
yjPy_j^PyjP: frequency of protein structure type jjj.
3. Their dynamics are:
xiR=kxkRfkR(R,P)QkiRRxiR\dot{x}_i^R = \sum_{k} x_k^R f_k^R(R, P) Q^R_{ki} - \phi^R x_i^RxiR=kxkRfkR(R,P)QkiRRxiR yjP=lylPflP(R,P)QljPPyjP\dot{y}_j^P = \sum_{l} y_l^P f_l^P(R, P) Q^P_{lj} - \phi^P y_j^PyjP=lylPflP(R,P)QljPPyjP
 where fRf^RfR and fPf^PfP are fitness functions interdependent on both RNA and protein states (from IV.B).
4. Coupling term.
Fitness of RNA depends on translation accuracy and stability provided by proteins, while fitness of proteins depends on coding fidelity and diversity provided by RNA. This coupling can be expressed as:
fiR=fiR(R)+gR(P)f_i^R = f_i^R(R) + \lambda \cdot g^R(P)fiR=fiR(R)+gR(P) fjP=fjP(P)+gP(R)f_j^P = f_j^P(P) + \mu \cdot g^P(R)fjP=fjP(P)+gP(R)
where \lambda and \mu are coupling coefficients controlling the strength of RNA--protein interdependence.
5. Emergent attractors.
The system converges toward coevolutionary attractors, stable states where RNA sequences and protein structures reinforce each other's persistence.
Trade-offs (from IV.B) prevent trivial optimization, ensuring multiple possible attractors (e.g., high-stability/low-diversity vs. low-stability/high-diversity regimes).
6. Agent-based extension.
For richer exploration, an agent-based model can represent individual RNA and protein variants interacting probabilistically.
Agents replicate, mutate, and form complexes; emergent properties such as modularity or ribosome-like assemblies arise from local interaction rules.
This coupled replicator--mutator formalism allows RNA and proteins to be modeled not as isolated populations, but as dynamically linked evolutionary agents. It provides a mathematical basis for the emergence of synchronized adaptation under coevolutionary constraints.
D. Coupling with predator--prey Lotka--Volterra extensions
