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Keywords:

  • control theory;
  • G matrix;
  • indirect genetic effect;
  • maternal effect;
  • quantitative genetics

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. 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 inline image, 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).

Materials and methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

A dual-inheritance model of phenotypic evolution

We assume that multiple traits (z) determine fitness, and that optimum fitness is achieved when the mean trait inline image is equal to some optimum inline image, which may vary from generation to generation.

To demonstrate the utility of the reachability matrix (Trentelmann et al., 2001) in evolutionary biology, we revisit a model of genetic and nongenetic inheritance first derived by Kirkpatrick and Lande (Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990):

  • display math(1)

Here, inline image is the change in mean phenotypes from one generation to the next, and β(t), t = 0, 1, 2 …, is a vector of linear selection gradients acting on these phenotypes. In this set-up, selection has acted on T + 1 generations. M is the matrix of maternal-effect coefficients (Kirkpatrick & Lande, 1989); element inline image is the effect of maternal trait j on offspring trait i. Each coefficient can be calculated using partial regression (Lande & Price, 1989; McGlothlin & Brodie III, 2009). When M = 0, the model reduces to strict genetic inheritance. inline image is the covariance between additive genetic effects and phenotype, often approximated under stationarity (Kirkpatrick & Lande, 1989) as inline image. inline image therefore captures direct effects of genes and indirect effects of genes expressed by parents (Kirkpatrick & Lande, 1989). E represents the environmental influences on the phenotype. G is the additive genetic variance-covariance matrix, and P the phenotypic variance-covariance matrix analogous to G. P is found by solving the matrix Lyapunov equation, which is a particular form of equation often found in dynamical systems, including control theory (Trentelmann et al., 2001) and evolutionary biology (Kirkpatrick & Lande, 1989):

  • display math

Here, ′ denotes matrix transpose, the matrices G, inline image, P and M are all n by n for the n traits under consideration. For discussion of the consequences of these assumptions in constructing this and similar models, see Hadfield (2012). The supporting information includes scripts for R (R Core Team, 2012) to run the model by numerically solving this Lyapunov equation.

Embedding quantitative genetics in a control systems framework

Control theory is concerned with dynamic systems that are influenced by external inputs and internal feedbacks (Trentelmann et al., 2001), which we take to be the genetic and nongenetic components. From a control systems perspective, eqn (1) is a multivariable, discrete-time, linear system with state inline image and multivariable inputs (here, generations of selection) β(t), t = 0, 1, …. To emphasize this point, we partition β(t) into Bu(t) and rewrite model 1 as

  • display math

where inline image and inline image. Here, inline image could be termed the indirect genetic path and inline image the genetic path. This partitioning of β(t) into B multiplied by u(t), where B is n×m and u(t) represents m independent parameters, allows us to explore and clarify the role of different selection regimes. For example, an optimum trait selection on trait 1, which fluctuates according to  sin ωt would be captured by β(t) = Bu(t) with inline image and u(t) =  sin ωt. Off-diagonal elements of B enable the modelling of a skewed selection and multiple environmental effects, such as slow (or weak) and fast (or strong) climatic oscillations.

Solving for inline image, t = 1, 2, …, T, we find (Appendix A) that the mean phenotype evolves over T + 1 generations of selection β(0), β(1), …,β(T), from inline image to

  • display math(2)

Here, inline image is the n by m × (T+1) block reachability matrix that has n rows (the number of traits under selection) and m(T + 1) columns determined by the T + 1 generations of m-dimensional selection and the cumulative build-up of maternal effects (Appendix A). inline image can be interpreted block by block:

  • display math

Selection on the phenotype at generation T makes contributions that accumulate generation by generation – the furthest-right block in inline image is the contribution of selection in the first generation (T = 0) to the trait at time T + 1 (i.e. the maternal effects from ‘generation’ 0); the furthest-left block is the contribution of selection in the inline image generation to the trait at time T + 1. A numerical example of how to calculate inline image is given in Table 1, and the supporting material contains scripts for R (R Core Team, 2012) that run the model. With each passing generation, the number of columns of inline image increases as the lags induced by the maternal effects accumulate.

Table 1. The ability of nongenetic inheritance to enable evolution into previously unreachable phenotypes depends on the interplay between genetic and nongenetic inheritance
M B T inline image Singular values UD Interpretation
  1. We now use three dimensions. G is inline image for all examples here. The environment and genetic variance favours all traits equally, meaning that any change in the distribution (orientation and shape) and dimensionality of ellipses of phenotypic change after selection is due to the maternal effects in M. The number of generations that selection has acted on is T + 1. The singular-value decomposition of inline image is inline image. D is a diagonal matrix containing the singular values, analogous to eigenvalues in an eigendecomposition. Similarly, U and V are made from corresponding right and left singular vectors (analogous to the corresponding right and left eigenvectors of G) respectively. The penultimate column of the table is UD and gives the evolvable directions (right singular vectors) scaled by the strength of selection (singular values) and so represents directions of evolution and the degree of resistance in that direction. The last column contains the biological interpretation of these patterns, for which we use the terminology of Houle (2001) and Mezey & Houle (2005).

inline image inline image 1 inline image inline image inline image Rotated ellipse (changed quantitative constraint)
2 inline image inline image inline image Not all phenotypes reachable (absolute constraint).
inline image inline image 1 inline image inline image inline image Rotated ellipse (changed quantitative constraint)
2 inline image inline image inline image Rotated ellipse (changed quantitative constraint).
inline image inline image 1 inline image inline image inline image Rotated ellipse (changed quantitative constraint)
2 inline image inline image inline image Not all phenotypes reachable (absolute constraint).
inline image inline image 1 inline image inline image inline image Rotated ellipse (changed quantitative constraint)
2 inline image inline image inline image Not all phenotypes reachable (absolute constraint).

The key component of inline image is the controlled part driven by selection β(t). When inline image, phenotypic evolution is governed by the linear span of the columns of inline image. The span of inline image determines whether genetic and nongenetic inheritance enable evolution into all phenotypic dimensions, regardless of where selection acts and any absolute genetic constraints. In a biological context, the span determines the independent phenotypic directions that can show a response to selection. The dimension of this span is the rank of inline image, which is given by the number of nonzero singular values of inline image. Any zero singular values therefore indicate absolute constraints on evolutionary change under genetic and nongenetic inheritance in an analogous way as zero eigenvalues of G under genetic inheritance (Blows & Hoffmann, 2005; Mezey & Houle, 2005).

Eigen- and singular-value decomposition

A simple, yet informative, approach to depicting how G and M interact to determine the distribution of responses to selection is to plot ellipsoids (Phillips & Arnold, 1989; Arnold et al., 2008), which envelop individual points to give a population-level measure of between-generation change (a simulation routine to demonstrate this is given in the supporting information). Under the multivariate breeders’ equation (Lande, 1979), these ellipses of the between-generation change are determined by the eigenvalues and eigenvectors of the symmetric matrix G (Phillips & Arnold, 1989). The response to selection, averaged over all possible patterns, is biased towards inline image, the dominant eigenvector of G and the direction of least genetic resistance (Schluter, 1996). As the dominance of the eigenvalue associated with inline image over all the other subdominant eigenvalues increases, so evolution is consequently biased more strongly towards inline image. This bias is encapsulated by the eccentricity of the ellipses of phenotypic change, which decreases as the strength of the evolutionary constraints increases (Blows & Hoffmann, 2005; Mezey & Houle, 2005; Blows, 2007). Note that by ‘eccentricity’ we mean the definition given by Jones et al. (2004), who defined the ‘eccentricity’ of an ellipse as the ratio of the smaller eigenvalue to the larger, and not the geometric definition of inline image, where a is the length of the ellipses's major axis and b the length of the ellipse's minor axis.

In general, inline image is neither square nor symmetric, so we cannot use an eigendecomposition of inline image to obtain information on the phenotypic variation after selection. Instead, we apply the more general singular-value decomposition. (For square symmetric matrices like G, the eigen- and singular-value decompositions are equivalent). The singular-value decomposition of inline image is inline image. D is a diagonal matrix containing the singular values, analogous to eigenvalues in an eigendecomposition. Similarly, U and V are made from corresponding right and left singular vectors (analogous to the corresponding right and left eigenvectors) respectively. The expected direction of least resistance with genetic and nongenetic inheritance is the dominant singular vector of inline image. By analogy with inline image (the dominant eigenvector of G), we call this inline image. inline image represents the direction of least resistance when genetic and nongenetic inheritance interact. If M is small, then inline image is approximately GB and M has only a minor effect on the direction of phenotypic change. However, if G and M are of similar magnitude, then the path of evolution with M will be very different to that without M.

We will explore the influence of various forms of M focusing on the sign patterns in the diagonal components (influences from mother to offspring in the same trait) and the sign and strength of off-diagonal components (influences from mother to offspring in different traits). Recall that the component element inline image in M is the effect of maternal phenotypic trait j on offspring phenotypic trait i ‘independently’ of genetic inheritance (Kirkpatrick & Lande, 1989). We assume that variation in the maternal traits has a heritable basis so that the influence of M plays out as an indirect genetic effect (Moore et al., 1997; McGlothlin & Brodie III, 2009). We first work with two traits and assume different M matrices:

  • display math

These matrices contain parameters over the range reported in a recent empirical review on maternal effects and inheritance (Räsänen & Kruuk, 2007). inline image and inline image represent, respectively, consistent positive or negative maternal investment in the corresponding offspring traits; inline image invokes a situation where the mother invests positively in trait 2 and negatively in trait 1. We vary the off-diagonal elements as stated in each case. When studying the impact of M on the number of phenotypes that can show a response to selection, we use three traits.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

The expected direction of evolution

We let the T + 1 generations of selection β(t) explore the unit sphere inline image uniformly. Therefore, no particular direction for divergence is preferred and, in the absence of genetic constraints and maternal effects, the ellipse of phenotypic change after T + 1 generations of selection would be spherical with radius inline image. Unless otherwise stated, we assume throughout that B = I, so that u(t) = β(t), and G = diag(1,0.4359). We consider four M matrices to illustrate how maternal effects interact with genetic constraints: inline image with inline image and inline image; inline image with inline image and inline image; inline image with inline image and inline image and inline image with inline image. For these four M matrices, we see that the maternal effects can maintain roughly the same expected evolutionary path (inline image and inline image, Fig. 1a,b) or change it significantly compared with the genetic-only path (inline image and inline image, Fig. 1c,d). The path is given by the direction of the major axis of the ellipse of phenotypic change, denoted as either inline image or inline image in the absence and presence of maternal effects respectively.

image

Figure 1. Evolutionary change with and without M (red and black respectively) under four different evolutionary scenarios over T = 2,3,4 and 5 generations of selection. For inline image (a) and inline image (b), the expected direction of evolution is relatively unaffected by nongenetic inheritance; for inline image (c) and inline image (d) the opposite is true: nongenetic inheritance rotates the expected path of evolution by between inline image and inline image. The eccentricity of the phenotypic ellipses is reduced by inline image (a) and inline image (c), so nongenetic inheritance relaxes evolutionary constraint; inline image (b) and inline image (d) impose stronger constraint on phenotypic evolution. The role of nongenetic inheritance in shaping the response to selection does not simply depend on positive or negative maternal effects, but also the nature of the correlations among the traits inherited. The matrices are inline image

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We next record the angle between the major axis of this ellipse with and without M after four generations of selection, that is, the rotation of the expected direction of least resistance caused by the maternal effects. To do this, we use inline image, inline image and inline image with off-diagonal parameters (inline image and inline image) varying between −0.4 and 0.4. If nongenetic inheritance is consistent in sign across both offspring traits, then the impact on the direction of evolution is often similar (Fig. 2a,b, but see Fig. 1c for an example of strong rotation with inline image). If the mother invests positively in one trait and negatively in another (e.g. inline image), then there is a strong rotation in the expected direction of evolution (Fig. 2c). The role of nongenetic inheritance in shaping the response to selection does not simply depend on positive or negative maternal effects, but also the nature of the correlations among the traits inherited (Figs 1 and 2).

image

Figure 2. Nongenetic inheritance can rotate the expected direction of evolution (top row, redder colours indicate a greater angle between inline image and inline image with the black contour at inline image) and change eccentricity of the ellipses of phenotypic change (bottom row, redder colours indicate more evolutionary constraint and the black contour indicates the genetic constraints determined by G only). An eccentricity (Jones et al., 2004) of 1 means that the two eigenvalues are the same and the ellipse is a circle. (a) & (d) use inline image (consistent positive maternal effects), (b) & (e) use inline image (consistent negative maternal effects) and (c) & (f) inline image (negative for trait 1 and positive for trait 2). The matrices used (with off-diagonal elements varying from −0.4 to 0.4) were : inline image

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Eccentricity and evolutionary constraint

We now focus on the effect that M has on the eccentricity of the ellipses of phenotypic change after selection. The form of M can constrain or unconstrain the multivariate shape of phenotypes after selection in each generation. Nongenetic inheritance can decrease (inline image and inline image, Fig. 1a,c) or increase (inline image and inline image, Fig. 1b,d) evolutionary constraint as measured by the eccentricity of the ellipses of phenotypic change. In this example where evolutionary constraint is increased, the spread along inline image is much greater (Fig. 1b). All four combinations of more/less constraint and same/changed direction of evolution are possible (Fig. 1) and depend on the form of M.

We next record the resulting eccentricity of the ellipses of phenotypic change after four generations of selection, that is, the effect on the quantitative constraints due to the maternal effects, using inline image, inline image and inline image with off-diagonal parameters (inline image and inline image) varying between −0.4 and 0.4. Negative inheritance (Fig. 2e) unconstrains phenotypic evolution more than positive inheritance (Fig. 2d), both across more of parameter space and also in relaxing of constraints (compare the greater blue area within the black contour in Fig. 2e over Fig. 2d). If the mother invests positively in one trait, but negatively in another (e.g. inline image), then we see high, but often slightly weaker, evolutionary constraints (Fig. 2f) accompanying the strong rotation in the expected direction of evolution discussed previously (Fig. 2c).

We next explore how the path of phenotypic evolution is affected by the strength of M and the distribution (orientation and shape) of variation around the population mean. We assume that selection pressure is periodic through time: β(t) = Bu(t), with inline image, that is, along inline image, and u(t) = cosπt/4 + ε(t), where ε(t) is normally distributed with mean 0 and standard deviation 0.1. This set-up represents a noisy, time-periodic selection pressure, which acts only on the trait with least evolutionary resistance. Choosing inline image, with inline image and inline image, the maternal matrix is small and the mean path deviates only slightly away from a straight line (Fig. 3a), as would be the case without maternal effects. However, with inline image and off-diagonal elements as above, that is, a four-fold increase in the strength of M up to the levels reported empirically, the path of evolution opens out and the changes in both traits are of similar magnitude (Fig. 3a). With each passing generation, M rotates the axes of ellipses of phenotypic change (see also Fig. 1c) and, despite selection on only one trait, weakens evolutionary constraint as the axes of the ellipses become more equal in length until what was the minor axis becomes the major axis, and vice versa (Fig. 3b). The ellipses of phenotypic change after selection are shaped by interaction of the genetic constraints, maternal (indirect genetic) effects and the small temporal uncertainty ε(t).

image

Figure 3. M affects the mean path of evolution and the shape of the ellipses of phenotypic change after selection in each generation. Selection pressure is assumed to be cyclic, following b(t) =  cos πt/4. The solid ellipses (a) and solid points (b) are for inline image with inline image, inline image and inline image. The dashed ellipses (a) and open symbols (b) are for inline image with the same off-diagonal elements. Lighter grey symbols are earlier generations (T = 0, 1, 2, …, 8); the filled ellipse in (a) is given using G as the initial conditions for both runs. An eccentricity (Jones et al., 2004) of 1 means that the two eigenvalues are the same and the ellipse is a circle.

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The dimensionality of evolution

We now consider three traits to investigate phenotypic evolution into more dimensions (Table 1). To understand the extent of the influence of M in reaching previously unreachable phenotypes, we can study the rank of M-dependent reachability matrix inline image, which can be determined by the number of zero singular values. In some cases, the space of achievable phenotypes with maternal effects contains those without maternal effects (i.e. is a strict superset). In this instance, the role of M is to change the eccentricity of the ellipses of phenotypic change after selection (Figs 1 and 2) and, in our examples, the role of M is to strengthen the evolutionary constraint because all the eigenvalues of G used in Table 1 are the same. This rotation is the case for inline image (Table 1) when selection acts on trait 2 over three generations or more, but not if selection acts on trait 1 for at least three generations (Table 1). When inline image (Table 1), not all phenotypes can show a response to selection. This example shows that the reachable phenotypes with M≠0 need not contain those without M = 0 because the maternal effects fail to overcome the genetic constraints. After three generations of selection on trait 1 with inline image, the quantitative and absolute constraints due to genetic and nongenetic inheritance are particularly clear: there is no variation in the third trait (column) and very little in the second trait (see UD of inline image in Table 1).

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

In the multivariate breeders' equation inline image (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 inline image (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 inline image 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 inline image.

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 inline image 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. inline image is necessarily larger than G to incorporate the cumulative, generation-lagged maternal effects. This singular-value decomposition of inline image 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, inline image 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) inline image. 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 inline image (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 inline image as an analytical descriptor of the consequences of genetic and nongenetic inheritance applies generally. A singular-value decomposition of inline image 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.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

We thank Angus Buckling, Jarrod Hadfield, Rebecca Hoyle, Rufus Johnstone, Bram Kuijper, Louise Machell, Jonathan Wells and two anonymous reviewers for insightful comments that improved earlier drafts. This work was supported by the Engineering and Physical Sciences Research Council (grant number EP/H031928/1).

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

Appendix A

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix A

The Maths

From the Kirkpatrick–Lande model (eqn (1); Kirkpatrick & Lande, 1989; Lande & Kirkpatrick, 1990), we have that

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Collecting terms in compact form we find, after T generations of selection, that

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Here β(t)=Bu(t), inline image and inline image. If we set

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then we have that

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U is the vector of selection pressures in the current (inline image) and all previous generations. inline image is an intermediate structure, similar in form to the reachability matrix, which we are coming to. Stacking the equations for inline image, we have that

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But, inline image. Hence

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For example, if T = 2, then

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In general,

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Note that we use inline image to denote matrix transpose, whereas Kirkpatrick & Lande (1989) used T. We use T in the sense of T generations. Note, finally, that for 1 ≤ k ≤ T − 1, the generation-lagged contribution that u(k) makes to inline image is given by

  • display math