To capture the ecological character of RNA--protein coevolution, we extend the replicator--mutator system with Lotka--Volterra--style coupling. This formalism introduces explicit interaction terms, treating RNA and protein populations as coevolving agents locked in a dynamic balance akin to predator--prey systems.
1. Basic Lotka--Volterra form.
The classical predator--prey equations are:
X=rXaXY\dot{X} = rX - aXYX=rXaXY Y=dY+bXY\dot{Y} = -dY + bXYY=dY+bXY
where XXX is prey, YYY is predator, rrr is prey growth rate, ddd is predator death rate, and a,ba, ba,b are interaction coefficients.
2. RNA--protein analogy.
RNA (RRR) supplies coding capacity, analogous to "prey."
Proteins (PPP) depend on RNA templates, but also stabilize RNA, analogous to "predators" that both consume and maintain prey.
The interaction is mutualistic but asymmetrical: proteins require RNA to exist, RNA requires proteins for stability.
3. The coupled equations become:
R=rRRP+RP\dot{R} = rR - \alpha RP + \eta RPR=rRRP+RP P=dP+RP\dot{P} = -dP + \beta RPP=dP+RP
where:
RP\alpha RPRP: cost of mismatch (translation errors, instability).
RP\eta RPRP: stabilizing effect of proteins on RNA (error correction, structural support).
RP\beta RPRP: benefit of coding templates for proteins.
4. Integration with replicator--mutator dynamics.
The Lotka--Volterra terms modulate effective fitness in the replicator--mutator equations (IV.C).
Thus, the growth rates of RNA and protein populations are shaped not only by intrinsic replication/mutation but also by cross-dependencies.
5. Emergent Red Queen dynamics.
The coupled system can exhibit oscillatory trajectories where RNA and protein populations "chase" each other in state space.
These cycles correspond to Red Queen dynamics: continual adaptation required just to maintain functional equilibrium.
In some parameter regimes, the system converges to a stable coexistence attractor (ribosome-like stability); in others, it oscillates or collapses (loss of synchrony).
6. Ecological interpretation.
