The genotype--phenotype mapping is not additive but shaped by epistasis (interaction between codons) and pleiotropy (one codon affecting multiple traits). Mathematically, the phenotype can be represented as: y=f(G;)+\mathbf{y} = f(G; \Theta) + \epsilony=f(G;)+
where y=(T,E,F,...)\mathbf{y} = (T, E, F, ... )y=(T,E,F,...) is the phenotype vector, fff is a nonlinear function parameterized by interaction matrix \Theta, and \epsilon is stochastic noise.
4. RNA motifs protein domains coupling.
Specific RNA motifs (e.g., stem-loops, codon clusters) correlate with conserved protein domains. This can be formalized as a bipartite graph B=(M,D,E)\mathcal{B} = (M, D, E)B=(M,D,E), where MMM is the set of RNA motifs, DDD is the set of protein domains, and EMDE \subset M \times DEMD encodes functional interactions.
The stability of mapping is given by the weight function:
1. w(mi,dj)=Pr(djmi)w(m_i, d_j) = \Pr(d_j \mid m_i)w(mi,dj)=Pr(djmi)
 representing the conditional probability that motif mim_imi reliably produces domain djd_jdj.
2. Emergent property: modularity.
This mapping naturally generates modular structures: clusters of RNA motifs co-map to clusters of protein domains, yielding functional modules.
Modularity acts as an attractor in CAS terms, stabilizing coevolution by buffering local mutations while preserving global function. Through this formalism, the genotype--phenotype map is modeled not as a static lookup table but as a dynamic, probabilistic, and feedback-driven system. This allows the exploration of how codon reassignments, dipeptide biases, and domain emergence coevolve in synchrony under selection pressures.
B. Interdependent fitness functions with trade-offs
In a coevolutionary system, the fitness of RNA and proteins cannot be defined independently. RNA provides coding capacity but depends on proteins for stability and catalysis; proteins provide structural and functional diversity but depend on RNA for inheritance and reproducibility. This creates interdependent fitness functions shaped by multiple trade-offs.
1. Joint fitness definition.
Let RRR denote the RNA component (e.g., codon usage profile, structural motifs) and PPP denote the protein component (e.g., folding stability, enzymatic efficiency). The joint fitness of the RNA--protein system can be formalized as:
W(R,P)=fR(R,P)+fP(R,P)C(R,P)W(R, P) = \alpha \cdot f_R(R, P) + \beta \cdot f_P(R, P) - \gamma \cdot C(R, P)W(R,P)=fR(R,P)+fP(R,P)C(R,P)
where:
fR(R,P)f_R(R, P)fR(R,P): RNA functionality conditional on protein support (e.g., translation fidelity, ribozyme--protein stability).
fP(R,P)f_P(R, P)fP(R,P): protein functionality conditional on RNA coding accuracy (e.g., folding efficiency, catalytic robustness).
C(R,P)C(R, P)C(R,P): coupling cost due to mismatch or inefficiency (e.g., unstable codon--amino acid assignment, misfolded peptides).
,,\alpha, \beta, \gamma,,: weighting coefficients reflecting ecological and evolutionary pressures.
