## Introduction

Localized gene dispersal is the major reason why species’ gene pools are not evenly mixed. The evolutionary consequences of this are profound, and can include divergence among populations and local adaptation, depending on the actions of natural selection and the vagaries of genetic drift (Slatkin 1985). Understanding of gene dispersal, and the factors restricting it, is therefore fundamental to evolutionary biology, and can also be applied to the genetic conservation of small or fragmented populations (Ellstrand 1992) and to the confinement of genetically modified (GM) organisms (Pilson & Prendeville 2004).

For many plant species, animal-mediated pollination is a principal mode of gene dispersal (Fenster 1991). Typically, it is highly localized (Bateman 1947; Schaal 1980; Fenster 1991) because of: the restricted movements of pollinators (Schmitt 1980); the rapid attenuation of flower-to-flower pollen carry-over (Morris *et al*. 1995); and the absence of secondary stigma-to-stigma transfers (Bateman 1947) due to pollen's apparent tendency to adhere to the first stigma on which it is deposited. Theoretically, gene dispersal can be predicted from pollinator movements and their associated patterns of flower-to-flower gene dispersal, or paternity shadow. Several studies have aimed to model patterns of pollinator-mediated gene flow (Bateman 1947; Schmitt 1980; Morris 1993) and, building on these, a recent theory known as the portion-dilution model (PDM) has made it possible to predict pollinator-mediated gene dispersal (Cresswell *et al*. 2002; Cresswell 2003, 2005). Use of the PDM requires knowledge of pollinator movements, which are in principle directly observable, and their associated paternity shadow (Fig. 1). However, determination of the paternity shadow under field conditions is problematic. Here we develop a feasible method to determine the paternity shadow relevant to field conditions where plants are exposed to ambient weather conditions and multiple visits by wild pollinators.

The PDM applies when a portion of a source population's paternity is realized in a sink population where this paternity is diluted among seeds of differing paternity (Fig. 1). The proportion of the sink's seed with source paternity, ξ, is given by:

where each pollinator arriving in the sink population from the source population fertilizes Ψ fruits with source paternity for every *b̄* flowers it pollinates, and where a proportion, *E*, of all the sink's pollinators arrive directly from the source (Cresswell *et al*. 2002). The parameters *E* and *b̄* relate to pollinator behaviour that can be observed directly. The parameter Ψ derives from the paternity shadow (Fig. 1). The experimental characterization of paternity shadows is therefore crucial to the quantitative prediction of gene dispersal by the PDM.

A paternity shadow can be quantified experimentally based on the proportions of genetically marked seeds in the unmarked flowers that a pollinator has visited after a visit to a single marked flower (Cresswell *et al*. 2002). However, these experiments are laborious and realistically feasible only under laboratory conditions, because they require a single pollinator first to visit a marked flower and then to visit consecutively a series of virgin, unmarked flowers, which is difficult to arrange in the field. Further complications arise if patterns of pollen transfer are influenced by variation in floral characteristics, such as levels of nectar and available pollen (Galen & Plowright 1985; Cresswell 1999). In this case, the relevant floral variables must be adjusted in the laboratory experiments to match field conditions if the emerging paternity shadow is to apply. We aimed to quantify the paternity shadow relevant to flowers under field conditions while avoiding complicated experimental procedures. The objectives of our study were to develop a theoretical framework for estimating paternity shadows from simple field experiments and to use this theory, together with data collected from the field, to estimate the paternity shadow of bumble bee pollinated oilseed rape (*Brassica napus*). *Brassica napus* is an economically important GM crop that is pollinated by both insects (Cresswell 1999) and wind (Eisikowitch 1981).

### overview of theoretical approach

Table 1 summarizes the notation used in the following exposition. Consider a pollinator that visits *W* flowers in a genetically marked population before moving to an unmarked population of the same plant species. The pollinator thereby transfers marked pollen that fertilizes some seeds at unmarked plants. What is the representation of marked seeds among the progeny of the unmarked plants? If *f*_{v} denotes the proportion of progeny fertilized by a particular flower at another flower that is visited *v* flowers later in a pollinator's foraging sequence, the collection of *f*_{v} for all *v* is the paternity shadow (Cresswell *et al*. 2002). Let *m* denote the maximum extent of the paternity shadow, i.e. *f*_{v} = 0 when *v* > *m*.

Variable | Definition |
---|---|

v | Sequential position of a component of the paternity shadow |

f_{v} | vth component of the paternity shadow |

m | Maximum extent of the paternity shadow, or number of its non-zero components |

W | Number of marked flowers visited initially by a pollinator |

i | Sequential position of an unmarked flower visited by pollinator after leaving marked flowers |

Φ_{i} | Proportion of marked seed in fruit of the ith unmarked flower visited by a pollinator after leaving marked flowers |

X | Spatial position of an unmarked plant |

P_{X,i} | Probability that a pollinator visiting a plant at X arrives on its ith flower visit after leaving the unmarked plants |

M_{X} | Proportion of marked seed set by a plant located at X |

n_{X,i} | Number of times pollinators were observed to visit a plant located at X on their ith flower visit after leaving marked flowers |

n_{X} | Total number of times pollinators were observed to visit a plant located at X |

α, β | Parameters governing the shape of the least-squares fitted paternity shadow |

(a)_{e} | Expected value of any variable, a, calculated from the fitted paternity shadow |

Suppose that pollen from the pollinator's visits to the marked plants fertilizes a proportion Φ_{i} of the fruit's seed at the *i*th unmarked flower that the pollinator visits. This proportion (Fig. 1) is compounded from the paternity shadows of the marked flowers, and is given (Cresswell 2005) by:

Let *P*_{X,i} denote the probability that a pollinator visiting the unmarked plant at location *X* arrives at its *i*th flower visit after leaving the marked plants. The *P*_{X,i} can be estimated from a collection of observations of pollinator movements by:

where *n*_{X,i} is the number of times pollinators visited a plant at location *X* on their *i*th flower visit after leaving the marked plants, and ∑*n*_{X,.} is the total number of visits observed at location *X*.

If *M*_{X} denotes the proportion of all seeds on a plant at location *X* that are marked, then, following Morris (1993):

or in matrix notation:

*=*

**M****,( eqn 5)**

*P*φwhere * M* is a vector containing the

*M*

_{X}, i.e.

*= {*

**M***M*

_{X}}. Eqn 4 applies to plants on which the flowers each receive one or more visits from pollinators (Cresswell 2003), provided that all the pollinators generate the same marked paternity shadows,

*f*

_{v}, and have patterns of movement governed by

*P*

_{X,i}.

If the number of plant locations is equal to the number of successive visits to unmarked flowers under consideration, denoted *r*, then * P* is a square matrix (

*r*×

*r*elements) with elements

*P*

_{X,i}, φ = {Φ

_{X}}, a vector of

*r*elements, and

*has*

**M***r*elements. If

*is non-singular, we can deduce φ by:*

**P****φ**=

**P**^{−1}

*.( eqn 6)*

**M**The paternity shadow can then be recovered from φ by equation 2 because:

_{i}− Φ

_{i+1}=

*f*−

_{i}*f*

_{W+ i}.( eqn 7)

Thus unique solutions for *f*_{v} are feasible provided that the number of plant locations exceeds *m*.

The preceding approach omits the possible contribution of sampling error to the observed values that must be used to populate * M* prior to solving equation 6. Therefore the approach is at risk of distorting the paternity shadow to account (mistakenly) for part of an observed pattern that is properly attributed to statistical noise. Least-squares regression analysis is a convenient tool for finding the best fit of a parametric model to data that contain sampling errors. We therefore developed a regression method to fit a biologically realistic, parametrically defined paternity shadow by using the matrix framework defined in equation 5. We required the paternity shadow to decrease monotonically in the form of one of the highly leptokurtic, parametric curves that fits well to patterns of flower-to-flower pollen transfer (Morris

*et al*. 1995). For simplicity, we chose the exponential power function (Cresswell 2005), although our calculations showed that fitting a Weibull function, as suggested by Morris

*et al*. (1995), gave almost identical results (data not shown). Thus the expected elements of the paternity shadow are given by:

*f*)

_{v}_{e}= exp(α

*v*

^{β})( eqn 8)

We obtained the expected elements of φ, denoted (φ_{i})_{e}, as functions of constants α and β using equations 2 and 8. Hence the expected value of * M*, which comprised the elements (

*M*

_{X})

_{e}, was calculated from (φ

_{i})

_{e}and

*using equation 5. We then implemented the least-squares regression technique by varying parameters α and β of eqn 8 to minimize the residual sum of squares (SSR) between the observed and expected values of*

**P***M*:

_{X}