Evolutionary branching under multi-dimensional evolutionary constraints

Abstract

The fitness of an existing phenotype and of a potential mutant should generally depend on the frequencies of other existing phenotypes. Adaptive evolution driven by such frequency-dependent fitness functions can be analyzed effectively using adaptive dynamics theory, assuming rare mutation and asexual reproduction. When possible mutations are restricted to certain directions due to developmental, physiological, or physical constraints, the resulting adaptive evolution may be restricted to subspaces (constraint surfaces) with fewer dimensionalities than the original trait spaces. To analyze such dynamics along constraint surfaces efficiently, we develop a Lagrange multiplier method in the framework of adaptive dynamics theory. On constraint surfaces of arbitrary dimensionalities described with equality constraints, our method efficiently finds local evolutionarily stable strategies, convergence stable points, and evolutionary branching points. We also derive the conditions for the existence of evolutionary branching points on constraint surfaces when the shapes of the surfaces can be chosen freely.

Evolutionary branching of function-valued traits under constraints

“…In adaptive dynamics theory, which is an extension of ESS theory for continuous strategies (Metz et al, 1996), directional evolution of function-valued traits as well as scalar and vector traits are described with an ordinary differential equation (Dieckmann and Law, 1996). Diversifying evolution of scalar and vector traits can be analyzed by examining evolutionary branching conditions (i.e., conditions for evolutionary singularity, strong convergence stability, and evolutionary instability) (Metz et al, 1996;Geritz et al, 1997Geritz et al, , 1998Dieckmann, 2012, 2014;Ito and Sasaki, 2016). As for function-valued traits, appropriate description of their evolution often requires constraints.…”

“…On the other hand, as for evolutionary branching conditions, while conditions for evolutionary singularity and evolutionary stability under constraints can be derived by the calculus of variations (e.g., Pontryagin's maximum principle) (Dieckmann et al, 2006;Parvinen et al, 2013;Metz et al, 2016), an analytically tractable method for examining convergence stability have not been developed yet. This paper develops conditions for strong convergence stability (Leimar, 2009) as well as evolutionary singularity and evolutionary stability, for function-valued traits under simple equality constraints, by extending the Lagrange multiplier method for vector traits (Ito and Sasaki, 2016) for infinite-dimensional vector traits. The set of conditions for evolutionary singularity, strong convergence stability, and evolutionary instability are called the candidate branching point conditions, or the CBP conditions.…”

“…The set of conditions for evolutionary singularity, strong convergence stability, and evolutionary instability are called the candidate branching point conditions, or the CBP conditions. The next section, Section 2, assumes a single constraint and applies the Lagrange multiplier method by Ito and Sasaki (2016) for discretized function-valued traits, and extends the discretized CBP conditions for function-valued traits. Section 3 derives the CBP conditions for function-valued traits under multiple constraints.…”

Selection for novel metabolic capabilities inSalmonella enterica

“…; Kvitek and Sherlock ; Salverda et al. ; Ito and Sasaki ). However, studies associating these constraints and trade‐offs with the evolution of novel phenotypes are limited (Downs ; Copley ; Chubiz and Marx ).…”

“…By themselves, the study of constraints and trade-offs accompanying adaptive evolution has been central in the field of evolutionary biology, both in eukary-otes (Mitchell-Olds 1996;Etterson and Shaw 2001;Ghalambor et al 2004;Shoval et al 2012) and in eubacteria (Mongold et al 1996;Bohannan et al 2002;Craig MacLean 2005;Duffy et al 2006;Gudelj et al 2007). A large number of studies have pointed out how these constraints and trade-offs can limit the spectrum of adaptive mutations and evolutionary trajectories (Weinreich et al 2005;Weinreich et al 2006;Kvitek and Sherlock 2011;Salverda et al 2011;Ito and Sasaki 2016). However, studies associating these constraints and trade-offs with the evolution of novel phenotypes are limited (Downs 2006;Copley 2012;Chubiz and Marx 2017).…”