## Introduction

Floral symmetry and its evolution have attracted much attention in plant evolutionary and developmental biology (Coen, 1996; Endress, 2001; Citerne *et al*., 2010). Evolution of floral symmetry and its developmental basis have been investigated both on a microevolutionary scale (Kim *et al*., 2008) and across large clades (Zhang *et al*., 2010; Bartlett & Specht, 2011; Howarth *et al*., 2011; Busch *et al*., 2012). Traditionally, flower morphology has been assessed in a qualitative manner, but increasingly investigators are quantifying floral shape and symmetry with the methods of geometric morphometrics (Shipunov & Bateman, 2005; Gómez *et al*., 2006; Frey *et al*., 2007; Benitez-Vieyra *et al*., 2009; Gómez & Perfectti, 2010; Nattero *et al*., 2010; van der Niet *et al*., 2010; Kaczorowski *et al*., 2012). Because floral symmetry is important as a potential target of selection and because it is key to understanding the development and evolution of flowers, morphometric analyses of floral shape need to identify the patterns of symmetric variation and asymmetry and quantify them. Many flowers show complex symmetry, with multiple components of asymmetry (e.g. left–right, adaxial–abaxial, or asymmetry under rotation), which are challenging for morphometric analysis but offer opportunities for innovative studies in different biological contexts. So far, however, no morphometric study of floral shape has used methods that explicitly take symmetry into account and can separate the different components of symmetric variation and asymmetry.

Morphometric analyses of floral symmetry and asymmetry for zygomorphic flowers are relatively straightforward, because methods for analysing objects with bilateral symmetry are firmly established (Auffray *et al*., 1999; Mardia *et al*., 2000; Kent & Mardia, 2001; Klingenberg *et al*., 2002). Nevertheless, these methods have not yet been used for the study of floral symmetry (but for applications to leaf asymmetry, see Albarrán-Lara *et al*., 2010; Klingenberg *et al*., 2012). For other types of floral symmetry, morphometric analyses are more difficult. Frey *et al*. (2007) proposed a measure of rotational symmetry based on how close a single set of corresponding landmarks is to a regular polygon. Symmetry and asymmetry of algal cells with two perpendicular axes of reflection symmetry were studied with a generalization of the approach for bilateral symmetry (Potapova & Hamilton, 2007; Savriama *et al*., 2010). Savriama & Klingenberg (2011) offered a further generalization that is applicable to any type of symmetry, which constitutes a general framework for morphometric studies of floral symmetry and asymmetry. This approach is based on the theory of symmetry groups, which offers a rigorous and flexible mathematical foundation for the analysis of symmetric shapes (Weyl, 1952; Armstrong, 1988; Savriama & Klingenberg, 2011). The method is a generalization of the geometric morphometric methods devised for the study of bilateral symmetry (Klingenberg & McIntyre, 1998; Mardia *et al*., 2000; Kent & Mardia, 2001; Klingenberg *et al*., 2002). Depending on the type of symmetry under study, this method can separate a component of symmetric variation from one or more components of asymmetric shape changes and it can further resolve the asymmetric components into directional and fluctuating asymmetry (Savriama & Klingenberg, 2011). This paper aims to bring this new method to the attention of plant biologists and to demonstrate it in a first application to the study of floral shape.

Fluctuating asymmetry has been widely used as an indicator of stress or individual quality, with variable results (Møller & Swaddle, 1997; Møller & Shykoff, 1999; Perfectti & Camacho, 1999; Roy & Stanton, 1999; Freeman *et al*., 2003b; Raz *et al*., 2011). Fluctuating asymmetry is usually thought to originate from random variation in the developmental processes that produce the structures of interest (Palmer & Strobeck, 1986; Klingenberg, 2003). It is therefore a component of within-individual variation (Herrera, 2009) and is of nongenetic origin, although genetic factors may modulate the expression of fluctuating asymmetry (Queitsch *et al*., 2002; Klingenberg, 2003; Leamy & Klingenberg, 2005; Takahashi *et al*., 2011). Because fluctuating asymmetry arises from variation in developmental processes, it is patterned by those processes and the analysis of the patterns of fluctuating asymmetry offers an opportunity to investigate the developmental origin of phenotypic variation (Klingenberg, 2010; Klingenberg *et al*., 2012).

The developmental processes that are responsible for establishing floral symmetry and asymmetry have been investigated with the tools of developmental genetics and are known in increasing detail (Coen, 1996; Endress, 2001; Citerne *et al*., 2010). These studies have revealed regulatory networks that play a major role in conferring adaxial or abaxial identities to floral organs (e.g. Coen, 1996; Almeida & Galego, 2005; Busch & Zachgo, 2007; Preston & Hileman, 2009; Citerne *et al*., 2010). Mutations of genes belonging to these networks can disrupt adaxial–abaxial polarity, but do not affect left–right symmetry (e.g. Preston & Hileman, 2009). By contrast, flowers with left–right asymmetry are relatively rare and no specific developmental pathways are known that establish this type of symmetry (Endress, 2001; Jesson & Barrett, 2002a; Marazzi & Endress, 2008). From this information, it appears that adaxial–abaxial and left–right asymmetries are the result of distinct sets of developmental processes, and therefore it is important to distinguish and characterize the patterns of fluctuating asymmetry for both components of asymmetry. The new morphometric methods for complex symmetry provide the tools for quantifying these components of fluctuating asymmetry and dissecting their patterns of variation.

We illustrate this approach by analysing symmetric and asymmetric components of floral shape variation in a population of *Erysimum mediohispanicum* grown under natural conditions (Gómez *et al*., 2006). As usual for the Brassicaceae, flowers of this species have four petals separated by two perpendicular axes of symmetry: one axis divides the petals of the left side from the right side and the other axis bisects the flower into adaxial and abaxial parts. Closer examination, however, reveals that there are many deviations from this ground plan (Fig. 1). A range of different symmetries occur even within single populations, there is a genetic basis for floral shape and symmetry, and different pollinators favour different floral symmetries (Gómez *et al*., 2006, 2008, 2009a,b,c; Gómez & Perfectti, 2010; Ortigosa & Gómez, 2010). Therefore, *Erysimum* flowers are a particularly interesting and relevant model for studying shape variation and the evolution of floral symmetry. Here we demonstrate the new morphometric approach to quantify the different components of symmetry and asymmetry and to investigate the patterns of variation associated with them. The case study demonstrates the utility of specifically quantifying symmetry and asymmetry for studies of floral morphology in evolutionary, functional and developmental perspectives (Breuker *et al*., 2006a; Klingenberg, 2010; Murren, 2012).