A mathematical framework for evo-devo dynamics

Abstract

Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Existing mathematical theory has considered either evolutionary dynamics while neglecting developmental dynamics, or developmental dynamics while neglecting evolutionary dynamics by assuming evolutionary equilibrium. We formulate a mathematical framework that integrates explicit developmental dynamics into evolutionary dynamics. We consider two types of traits: genetic traits called control variables and developed traits called state variables. Developed traits are constructed over ontogeny according to a developmental map of ontogenetically prior traits and the social and non-social environment. We obtain general equations describing the evolutionary-developmental (evo-devo) dynamics. These equations can be arranged in a layered structure called the evo-devo process, where five elementary components generate all equations including those describing genetic covariation and the evo-devo dynamics. These equations recover Lande's equation as a special case and describe the evolution of Lande's G-matrix from the evolution of the phenotype, environment, and mutational covariation. This shows that genetic variation is necessarily absent in some directions of phenotype space if at least one trait develops and enough traits are included in the analysis so as to guarantee dynamic sufficiency. Consequently, directional selection alone is generally insufficient to identify evolutionary equilibria. Instead, "total genetic selection" is sufficient to identify evolutionary equilibria if mutational variation exists in all directions of control space and exogenous plastic response vanishes. Developmental and environmental constraints influence the evolutionary equilibria and determine the admissible evolutionary trajectory. These results show that development has major evolutionary effects.

How development affects evolution

“…Assume for now that there is (i) no niche construction ( ∂ ε ⊤ / ∂ z = 0 ), (ii) no social development , (iii) no exogenous environmental change ( ∂ ε /∂τ = 0 ), and (iv) no direct effects of gene expression on fitness ( ∂w/∂ y = 0 ) (for an explanation of our use of matrix calculus notation, see Supplementary Information section 1). Then, González-Forero and Gardner (2021) show that the (expected) evolutionary dynamics of the phenotype satisfy provided that the developmental constraint in Eq. (1) is satisfied at all times (derivatives are evaluated at throughout; we use an equal sign in Eq.…”

“…For a-b , (from Layer 5, Eq. 2a in González-Forero and Gardner 2021). Similarly, for c-d , , with d z ⊤ /d y and s x /s y evaluated at the socio-devo equilibrium x ** that solves x ** = g ( y,x **).…”

“…Evolutionary dynamic equations in gradient form for developed traits (i.e., with explicit developmental constraints) such as in Eq. (3) only became available recently (González-Forero and Gardner, 2021). This makes it possible to analyse how the evolution of developed traits proceeds on the fitness landscape.…”

“…The singularity of G z means that the evolutionary dynamics are strongly non-standard;in particular, consideration of direct directional selection alone ( ∂w/∂ z ) is not sufficient for predicting possible evolutionary outcomes. Although direct directional genetic selection ( ∂w/∂ y ) is traditionally assumed to be zero in quantitative genetics, life history models with explicit developmental dynamics show this is not necessarily the case (see example in González-Forero and Gardner, 2021, where gene expression modulates resource allocation to fertility and is thus under direct selection in a classic life history model). In any case, the need to use the extended Lande equation (7) persists as absence of direct directional genetic selection does not necessarily eliminate the dependence of the right-hand side of Eq.…”

“…We say a gene expression, phenotype, and environment m ** = ( x **; y **; ϵ **) is a sociodevo equilibrium if and only if an individual with such gene expression and environment developing in the context of individuals having such gene expression, phenotype, and environment develops their same phenotype. A socio-devo equilibrium is socio-devo stable if and only if socio-devo stabilization dynamics are locally stable to small perturbations in the phenotype (González-Forero and Gardner, 2021). Although social development can yield socio-devo unstable phenotypes, our analyses apply only to socio-devo stable phenotypes (which are guaranteed to be so if all the eigenvalues of have absolute value strictly less than one, which we assume).…”