### Abstract

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Appendix A

The genetic variance-covariance (**G**) matrix describes the variances and covariances of genetic traits under strict genetic inheritance. Genetically expressed traits often influence trait expression in another via nongenetic forms of transmission and inheritance, however. The importance of non-genetic influences on phenotypic evolution is increasingly clear, but how genetic and nongenetic inheritance interact to determine the response to selection is not well understood. Here, we use the ‘reachability matrix’ – a key analytical tool of geometric control theory – to integrate both forms of inheritance, capturing how the consequences of generation-lagged maternal effects accumulate. Building on the classic Lande and Kirkpatrick model that showed how nongenetic (maternal) inheritance fundamentally alters the expected path of phenotypic evolution, we make novel inferences through decomposition of the reachability matrix. In particular, we quantify how nongenetic inheritance affects the distribution (orientation and shape) of ellipses of phenotypic change and how these distributions influence subsequent evolution. This interweaving of phenotypic means and variances accumulates generation by generation and is described analytically by the reachability matrix, which acts as an analogue of **G** when genetic and nongenetic inheritance both act.

### Introduction

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Appendix A

The genetic variance–covariance matrix **G** (Lande, 1979) embodies the genetic correlations among phenotypes. It has been studied intensively (Arnold *et al*., 2008) because it neatly encapsulates the response to selection according to the multivariate breeders’ equation (Lande, 1979). One measure of how genetic correlatrions constrain adaptation was given by Arnold (1992), who defined ‘genetic constraints’ as the pattern of genetic variation and covariation by taking the eigendecomposition (also known as the spectral decomposition) of **G**. This decomposition has become fundamental in our understanding of multivariate selection pressures (Blows, 2007) because it elegantly captures the predicted direction of evolution (given by , the dominant eigenvector of **G**, Schluter, 1996), the distribution (orientation and shape) of phenotypes in multivariate space (Phillips & Arnold, 1989; Jones *et al*., 2004) and the combinations of phenotypes for which variation exists (Blows & Hoffmann, 2005; Mezey & Houle, 2005). These facets represent different types of constraints on unrestricted phenotypic evolution. Houle (2001) and Mezey & Houle (2005) drew the distinction between quantitative and absolute constraint. Quantitative constraints are determined by the distribution (orientation and shape) of variation in multivariate space (given by the eigenvalues associated with each eigenvector). Absolute constraint occurs when **G** has at least one zero eigenvalue and is therefore not of full rank, which means there is no variation in the corresponding combinations of phenotypes (Blows & Hoffmann, 2005; Mezey & Houle, 2005). Evolution therefore cannot generate a response to selection in all phenotypic combinations. In summary, an eigendecomposition of **G** provides a comprehensive description of phenotypic evolution according to genetic (Mendelian) inheritance and the multivariate breeder's equation (Lande, 1979).

When additional nongenetic inheritance systems act, predictions based solely on **G** are incomplete because they neglect nongenetic or indirect genetic inputs to the phenotype (Kirkpatrick & Lande, 1989; Moore *et al*., 1997; Jablonka & Lamb, 2005; Bonduriansky & Day, 2009). Nongenetic inheritance occurs when a genetically expressed trait in one individual influences trait expression in another. The empirical evidence for the role of nongenetic inheritance in phenotypic evolution is increasingly comprehensive (Jablonka & Raz, 2009; Bonduriansky *et al*., 2012) and, of its many forms (Jablonka & Lamb, 2005; Day & Bonduriansky, 2011), parental effects are especially important for evolution because they map the consequences of individual responses to environmental stimuli transgenerationally (Badyaev, 2009). In particular, maternal effects describe how mothers influence their offspring's development and life-history (Falconer, 1965; Mousseau & Fox, 1998; Räsänen & Kruuk, 2007). We study maternal effects via the ‘maternal effect coefficient’ (Kirkpatrick & Lande, 1989), which parameterizes the influence of the mother's phenotype on the offspring's phenotype, ‘independent’ of inherited genes (Falconer, 1965; Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990; Hadfield, 2012). Assuming a heritable basis to variation in the maternal traits, this is an example of an indirect genetic effect (Moore *et al*., 1997; Wolf & Brodie III, 1998; McGlothlin & Brodie III, 2009; Bijma, 2011), because the maternal effect is determined by the mother's phenotype, which has a genetic component.

One consequence of this maternal effect is evolutionary momentum (Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990): even if selection stops, evolution continues because of inertia (memory) from the lag between maternal and offspring phenotypes. The lag is one reason why nongenetic inheritance can provoke evolutionary dynamics that do not arise under strict Mendelian inheritance (Day & Bonduriansky, 2011). Another fundamental change in the patterns of phenotypic evolution is how nongenetic inheritance can cause evolution in directions that are otherwise inaccessible (Kirkpatrick & Lande, 1989; Day & Bonduriansky, 2011). Kirkpatrick & Lande (1989) and Moore *et al*. (1997) showed how phenotypic change can occur despite an absence of additive genetic variance underpinning a particular trait. One example is to imagine maternal performance for increasing offspring body size (assumed as heritable and under selection), but with no direct effect of offspring genes on their own body size. Offspring body size would still evolve as a correlated response to selection on maternal performance despite the relevant elements of **G** being zero. While there is plenty of documentation on how maternal effects shape the mean path of evolution and the ability to circumvent absolute genetic constraints (Kirkpatrick & Lande, 1989; Moore *et al*., 1997), no attention – to our knowledge – has been paid to how it affects the distribution (orientation and shape) of between-generation responses to selection.

Here, we embed the dual-inheritance model of Kirkpatrick & Lande (1989) in a control systems framework, which allows us to focus on how the consequences of genetic and nongenetic inheritance become increasingly interwoven with each passing generation. This interweaving of **G** and **M** plays out in three principal ways, all of which are captured by a singular value decomposition of the ‘reachability matrix’ (a key analytical tool in geometric control theory Trentelmann *et al*., 2001), by changing (1) the expected direction of phenotypic evolution; (2) the distribution (orientation and shape) of between-generation responses to selection; and (3) the dimensionality of the phenotypes under selection. We introduce the reachability matrix to evolutionary biology here, and show how its role, under dual inheritance, is analogous to that of **G** under genetic inheritance (Lande, 1979).

### Discussion

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Appendix A

In the multivariate breeders' equation (Lande, 1979), the direction of least genetic resistance (and overall orientation), shape of the ellipses of phenotypic change after selection and the dimensionality of the response to selection are determined, respectively, by the eigenvectors, eigenvalues and rank of **G**. With nongenetic inheritance, **G** is insufficient to predict short-term evolutionary responses (Kirkpatrick & Lande, 1989; Jablonka & Lamb, 2005; Day & Bonduriansky, 2011). Using tools from geometric control theory, we introduce the reachability matrix (Trentelmann *et al*., 2001) to act as the analogue of **G** in a model of multivariate phenotypic evolution by genetic and nongenetic inheritance. The matrix is made up of blocks, each of which captures the transgenerational effect that selection on generation *T* has on the mean trait at generation *T* + 1. In a dual-inheritance model, the direction of least genetic resistance (and overall orientation), shape of the ellipses of phenotypic change after selection and the dimensionality of the response to selection are determined, respectively, by the singular vectors, singular values and rank of .

The classic dual-inheritance model of Kirkpatrick and Lande (Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990) revealed ways in which genetic and nongenetic inheritance fundamentally alter evolutionary dynamics, including evolutionary momentum and a way for traits that lack genetic variation to evolve (see also Moore *et al*., 1997). Using singular value decomposition, we extend these arguments in a coherent analytical framework. For example, the rank of lets us track whether indirect genetic effects (Moore *et al*., 1997) or maternal inheritance (Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990) circumvent absolute genetic constraints (Blows & Hoffmann, 2005; Mezey & Houle, 2005). As with the closely related eigendecomposition, the singular-value decomposition involves finding basis directions, but it has greater generality as the matrix under consideration need be neither square nor symmetric. When nongenetic inheritance acts, the use of eigenvalues and eigenvectors to predict evolutionary trends is inadequate because selection pressure is not described by the square, symmetric **G** matrix. is necessarily larger than **G** to incorporate the cumulative, generation-lagged maternal effects. This singular-value decomposition of lets us ask questions that Kirkpatrick & Lande (1989) could not, such as how the eccentricity of the ellipses of phenotypic change after selection interacts with genetic constraints. When there are no maternal effects and selection acts equally on all traits, is essentially *T* copies of **G** so that the singular values and vectors are the same as the eigenvalues and eigenvectors. Evolutionary paths with weaker (or zero) maternal effects oscillates back and forth near (or along) . As the maternal effects become stronger, this tracking ‘spins out’ to expand the reach of phenotypic evolution (Fig. 3a) and can weaken evolutionary constraint within each generation to the extent that what was the major axis of the ellipses of phenotypic change becomes the minor axis, and vice versa (Fig. 3b).

Our control theory approach emphasizes two critical points about dual inheritance: genetic and nongenetic inheritance do not combine additively from one generation to the next (Appendix A; we assume additive genetic effects *sensu* Fisher, 1918) and this interweaving builds up through the cumulative effect of selection pressure in and across previous generations (eqn (2)). The ability of the individuals in a population to evolve is not dependent simply on the additive set of possible trait values spanned by **G** and **M** (Appendix A) because the responses to selection accumulate and interweave with each passing generation. Selection in the earlier generation leaves a long-lasting signature on phenotypic evolution. The changing size, shape and orientation of the phenotypic ellipses after selection (Figs 1 and 2) and the increasing interweaving of genetic and nongenetic inheritance with each passing generation (Figs 1 and 3) show how the impact of dual inheritance is more than just adding momentum (Kirkpatrick & Lande, 1989; Day & Bonduriansky, 2011) or extending the dimensionality (rank) of phenotypic evolution (Kirkpatrick & Lande, 1989; Moore *et al*., 1997). The major advantage of the control theory approach we advocate here is an analytical descriptor of how genetic and nongenetic inheritance combine to narrow the ellipses of phenotypic change, as often occurs when maternal effects are positive or negative in both traits (Fig. 2d,e). If, on the other hand, maternal effects are positive on one trait and negative on another, two effects are particularly clear: (1) the expected direction of evolution is rotated more dramatically (Fig. 2c) and (2) the eccentricity of the trait ellipses increases and so biases evolution less tightly along (Fig. 2f). This impact of maternal effects on the phenotypic variance in the offspring generation has received less attention than impacts on the phenotypic means, but incorporating it in analysis has important consequences for predictions of phenotypic evolution. Wolf & Brodie III (1998) showed how stabilizing selection on a trait influenced by maternal effects favours a genetic correlation among direct genetic and indirect maternal effects that is opposite in sign to the maternal effect coefficient. Hoyle & Ezard (2012), using a univariate model of adaptation via maternal effects, phenotypic plasticity and an additive genetic component, showed how negative maternal effects minimize phenotypic variance to maximize fitness in relatively stable environments by keeping the population mean phenotype closer to its optimum.

Geneticists and evolutionary biologists have long used matrix algebra to reveal the influence of **G** (Lande, 1979; Blows, 2007). Here, we use geometric control theory tools to reveal the interwoven influences of **G** and **M**. The clear evolutionary biology interpretation of these control theory objects and the established geometric control theory for complex systems (Bloch, 2003) suggests greater scope for applying these concepts to more elaborate models of evolution. These concepts could, for example, decompose spaces of selection (inputs) and phenotype (outputs) according to the strength of stabilizing selection, assess the impact of inheritance via other relatives or help understand how **G** evolves under the influence of **M**, or vice versa. While we restrict ourselves to a single model (Kirkpatrick & Lande, 1989), the usefulness of the reachability matrix as an analytical descriptor of the consequences of genetic and nongenetic inheritance applies generally. A singular-value decomposition of unpicks the effects of the interaction of genetic and nongenetic inheritance mediated via maternal effects on the means (Fig. 3), distributions (shape and orientation, Figs 1 and 3) and rank (Table 1) of the phenotypes after selection when genetic and nongenetic inheritance interact.