  The relationship between genotype and phenotype plays a crucial role in
determining the function and robustness of biological systems. Here the
evolution progresses through the change in genotype, whereas the selection is
based on the phenotype, and genotype-phenotype relation also evolves. Theory
for such phenotypic evolution remains poorly-developed, in contrast to
evolution under the fitness landscape determined by genotypes. Here we provide
statistical-physics formulation of this problem by introducing replicas for
genotype and phenotype. We apply it to an evolution model, in which phenotypes
are given by spin configurations; genotypes are interaction matrix for spins to
give the Hamiltonian, and the fitness depends only on the configuration of a
subset of spins called target. We describe the interplay between the genetic
variations and phenotypic variances by noise in this model by our new approach
that extends the replica theory for spin-glasses to include spin-replica for
phenotypes and coupling-replica for genotypes. Within this framework we obtain
a phase diagram of the evolved phenotypes against the noise and selection
pressure, where each phase is distinguished by the fitness and overlaps for
genotypes and phenotypes. Among the phases, robust fitted phase, relevant to
biological evolution, is achieved under the intermediate level of noise
(temperature), where robustness to noise and to genetic mutation are
correlated, as a result of replica symmetry. We also find a trade-off between
maintaining a high fitness level of phenotype and acquiring a robust pattern of
genes as well as the dependence of this trade-off on the ratio between the size
of the functional (target) part to that of the remaining non-functional
(non-target) one. The selection pressure needed to achieve high fitness
increases with the fraction of target spins.
