Dimensional Reduction in Evolving Spin-Glass Model: Correlation of Phenotypic Responses to Environmental and Mutational Changes

The evolution of high-dimensional phenotypes is investigated using a statistical physics model consisting of interacting spins, in which phenotypes, genotypes, and environments are represented by spin configurations, interaction matrices, and external fields, respectively. We found that phenotypic changes upon diverse environmental change and genetic variation are highly correlated across all spins, consistent with recent experimental observations of biological systems. The dimension reduction in phenotypic changes is shown to be a result of the evolution of the robustness to thermal noise, achieved at the replica symmetric phase.

Figure 1

(a) $T$ dependence of the averaged fitness. (b) Fraction of matrices $\mathcal{J}(T)$ in which the BP algorithm does not converge within ${10}^5$ steps. For (a) and (b), the vertical dashed lines indicate the phase transition temperatures. (c)–(d) Scatter plots of ${\chi }_{i1}$ and ${\chi }_{i2}$ at (c) $T=1$ and (d) $T=0.4$. The slope of the diagonal line in (c) is 1. (e) $T$ dependence of the averaged correlation coefficients between $\{{\chi }_{i1}\}\cdots \{{\chi }_{i,N_T}\}$. The statistical errors over $\mathcal{J}(T)$ is smaller than the point size in the figure for all $T$ region, and error bars are not discernible.

Figure 2

Relationship between $\{{\chi }_i\}$ and $\{{\mathcal{M}}_i\}$ for one evolved genotype $\mathit{J}$ at (a) $T=1$ (RS) and (b) $T=0.2$ (RSB). These behaviors are commonly observed for any evolved $\mathcal{J}(T)$.

Figure 3

$T$ dependence of (a) averaged first and second eigenvalues of $\mathit{J}$, (b) correlation coefficient between $\mathrm{arc}\tanh ({\mu }_i)$ and ${\xi }_i^1$, and (c) averaged $d$. Vertical dashed line denotes transition temperatures. (d) $\epsilon$ dependence of $d$ for $T=1$ (RS) and $T=0.5$ (RSB).