2. Trade-offs in adaptation.
Thermostability vs. flexibility: proteins that are highly stable may sacrifice catalytic adaptability; RNA motifs that are rigid may sacrifice coding efficiency.
Exploration vs. exploitation: high RNA mutability enhances exploration of coding space but risks destabilizing established protein domains; protein conservatism stabilizes function but reduces adaptability.
Short-term vs. long-term fitness: an RNA mutation may immediately destabilize codon--amino acid mapping but enable later innovations in protein structure.
3. Context dependence.
Fitness functions are ecologically embedded. For example, under high thermal stress, thermostability terms dominate; in nutrient-limited environments, catalytic efficiency is weighted more heavily.
This context sensitivity can be modeled by letting coefficients ,,\alpha, \beta, \gamma,, vary dynamically with environmental parameters E(t)E(t)E(t).
4. W(R,P;E(t))=(E(t))fR+(E(t))fP(E(t))CW(R, P; E(t)) = \alpha(E(t)) f_R + \beta(E(t)) f_P - \gamma(E(t)) CW(R,P;E(t))=(E(t))fR+(E(t))fP(E(t))C
5. Stability conditions.
The RNA--protein system stabilizes when the joint fitness W(R,P)W(R, P)W(R,P) reaches a local maximum, corresponding to a coevolutionary attractor.
Trade-offs ensure that no single component maximizes fitness in isolation; instead, balanced optimization across RNA and protein is required.
This interdependent fitness formalism reframes adaptation not as an optimization of isolated entities, but as a coevolutionary negotiation constrained by trade-offs, feedback, and ecological pressures. It provides the foundation for embedding RNA--protein dynamics into replicator--mutator and predator--prey style models.
C. Replicator--mutator dynamics for coupled populations
To capture the coevolution of RNA and proteins, we extend the replicator--mutator framework to model two coupled populations: RNA sequences (RRR) and protein structures (PPP). Each population evolves under mutation, selection, and feedback from the other.
1. Classical replicator--mutator equation.
For a population of types i=1,...,ni = 1, \ldots, ni=1,...,n, the replicator--mutator equation is:
xi=j=1nxjfjQjixi\dot{x}_i = \sum_{j=1}^n x_j f_j Q_{ji} - \phi x_ixi=j=1nxjfjQjixi
where:
