Department of Chemistry, The University of Chicago, 929 E 57th St, Chicago, IL 60637, USA

James Franck Institute, The University of Chicago, 929 E 57th St, Chicago, IL 60637, USA

Department of Molecular Genetics and Cell Biology, CLSC 1106, 920 E 58th St, Chicago, IL 60637, USA

Institute for Biophysical Dynamics, The University of Chicago, 929 E 57th Street, Chicago, IL 60637, USA

Department of Physics, The University of Chicago, 5740 S Ellis Ave, Chicago, IL 60637, USA

Department of Biology, Knox College, 2 E South St, Galesburg, IL 61401-4999, USA

International Institute of Information Technology, Gachibowli, Hyderabad 500 032, Andhra Pradesh, India

Abstract

Background

Plant biologists have long speculated about the mechanisms that guide pollen tubes to ovules. Although there is now evidence that ovules emit a diffusible attractant, little is known about how this attractant mediates interactions between the pollen tube and the ovules.

Results

We employ a semi-

Conclusions

We propose a stochastic model that captures these dynamics. In the model, a pollen tube senses a difference in the fraction of receptors bound to an attractant and changes its direction of growth in response; the attractant is continuously released from ovules and spreads isotropically on the medium. The model suggests that the observed slowing greatly enhances the ability of pollen tubes to successfully target ovules. The relation of the results to guidance

Background

In flowering plants, unlike animals, the male and female germ units are multicellular, haploid structures that develop in different organs of the flower (Fig.

Schematics of fertilization

**Schematics of fertilization in vivo and in vitro**. (A) Schematic depiction of the pollen tube path through the ovary. Dashed box shows growth between the rows of ovules after emergence in the ovary chamber. pg-pollen grain, pt-pollen tube, si-stigma, st-style, oc-ovary chamber, ov-ovules. (B) Schematic depiction of an ovule and a pollen tube approaching the micropyle.

Many mechanisms have been proposed to explain how pollen tubes are guided to ovules, including mechanical tracts that direct growth, surface-expressed guidance cues, and diffusing signals

A series of semi-

Here we present a quantitative analysis of newly obtained time-lapse images from such a semi-

To explore the implications of these results, we developed a mathematical model of pollen tube response to a gradient of a diffusible attractant that is continuously released by the ovules. Because little is known about the receptors and internal signals that drive pollen tube response to such attractants, our model makes no assumptions about the molecular mechanism for sensing this gradient and instead focuses on whole-cell features, an approach which has been used to model algae phototaxis

Results

Incubation time influences pollen tube response

Previous semi-

Dissected ovules from

Experimental details

**heat-treated**

**0 hours**

**2 hours**

**4 hours**

Timing (hours)

Stigma

0

0

0

0

Pollen

0

0

0

2

Ovules

0

2

0

0

Imaging

2.5

2.5

2.5

4.5

Count (number)

Stigmas

4

7

5

5

Ovules

Penetrated/total

0/12

14/21

12/15

15/15

Pollen tubes

149

223

175

132

Starting distance

(

393.57 ± 17.85

415.68 ± 16.62

384.36 ± 20.26

430.16 ± 20.19

For various incubation times, these were the relative times that the stigmas were placed on the medium, the ovules were placed on the medium, the stigmas were pollinated, and imaging was started. Stigmas were always placed at 0 hours. The time for placing the ovules and for pollinating the stigmas was adjusted to give the ovules additional incubation time on the medium. Pollen tubes emerged 2-2.5 hours after pollination. For each incubation time, the total number of stigmas sampled, ovules penetrated and number pollen tubes analyzed is listed. The starting distance reported is the average distance between the center of the transmitting tract and the micropyles. These distances were not significantly different from each other (

To assay the amount of attraction that pollen tubes had toward an ovule, we calculated the fraction of pollen tube tips that were within a certain distance of a micropyle that grew either closer to (_{
closer
}) or farther from (_{
farther
}) that micropyle by the next time point (Δ_{
total
}) and how many of these tips had moved into either a closer bin (_{
closer
}) or a farther bin (_{
farther
}) at time _{
closer
}= _{
closer
}/_{
total
}and _{
farther
}= _{
farther
}/_{
total
}.

Pollen tube attraction to ovules

**Pollen tube attraction to ovules**. (A) Ovules and the cut stigma (upper left corner) are shown in red. Pollen tubes are shown, emerging from the style, in blue. The white concentric circles depict radial bins of 50, 100, 150, 200 _{mp }and _{tip }angles used in the analysis of pollen tubes turning. The _{mp }angle indicates how much the pollen tube would have to turn to take the most direct path toward the micropyle. The _{tip }angle describes the new direction chosen by the pollen tube in response to the gradient. (D) Circular standard deviations σ_{0 }for distributions of Δ_{mp }≥ 0). The key for the bars shown in B and D is the same.

Using this approach, we examined these frequencies for ovules that had incubated on the medium for 0, 2, and 4 hours. As a negative control, we used heat-treated ovules that had been incubated for 2 hours. This incubation time was chosen to be consistent with previous experiments in _{
farther
}) from the micropyle of ovules that had been incubated for 4 hours was significantly different (_{
farther
}, the effects of incubation time on _{
closer
}were less visible (Fig.

The previous statistics include pollen tube growth that occurs both before and after the pollen tube penetrates the ovule. The points after penetration were included to allow an unbiased comparison with the heat-treated control but may affect the trends in _{
closer
}and _{
farther
}. To prevent polyspermy, the interactions between pollen tubes and ovules change once an ovule is fertilized, which occurs shortly after pollen tube penetration _{
farther
}correspond to times before the nearest ovule was penetrated. The trends in these frequencies were consistent with those reported above for _{
closer
}and _{
farther
}, although the differences between the three incubation times in

Effectively, _{
farther
}quantifies the degree to which ovules cause pollen tubes to deviate from random growth once they come within a certain distance of a micropyle, but this statistic does not address whether growth while approaching this region is directed. To further analyze pollen tube approach, we defined two angles: _{
mp
}and _{
tip
}. The angle _{
tip
}is the angle that a pollen tube turns as it grows, and the angle _{
mp
}is the angle that a pollen tube would have to turn to grow directly toward the micropyle (Fig. _{
mp
}- _{
tip
}, measures how much pollen tube growth deviates from the most direct path toward the micropyle (Δ_{
mp
}≥ 0.

We use these statistics to summarize how different incubation times affected the deviation in guidance represented by the Δ_{0 }decreased with the incubation time, and the heat-treated control had the widest distribution (Fig. _{0 }of the functional ovules, but all three were significantly different (

In each experiment, the pollen tubes grew similar distances before reaching the ovules, which indicates that the difference in response results from the ovule incubation time. These data support a model where ovules release a diffusible signal (attractant) throughout the experiment, independently of the presence of pollen tubes. The data also suggest a putative range over which the response operates: both the frequency _{
closer
}and the distribution of the angle Δ

The pollen tube response is consistent with following a gradient

Previous studies have focused their analysis on only the sharp, obvious turns that pollen tubes make near the micropyle, both _{
tip
}and _{
mp
}in Fig. _{
mp
}. In _{
tip
}is the magnitude of the gradient at a tip, Δ_{
tip
}sin _{
mp
}(Fig. _{
tip
}is in units of the change of concentration per unit distance. If a pollen tube is following a gradient of attractant, then its turns should be correlated with Δ_{
mp
}.

Pollen tube behavior is consistent with turning in response to a gradient of an attractant across the tip surface

**Pollen tube behavior is consistent with turning in response to a gradient of an attractant across the tip surface**. (A) Schematic of gradient-following model. The pollen tube tip is treated as flat. A gradient in the attractant (_{tip}) concentration gives a difference in concentration Δ_{tip }= _{mp }+

To test this hypothesis, we looked at the relation between _{
tip
}and sin_{
mp
}by fitting the line _{
tip
}= _{
mp
}+

Pollen tube turning responses.

**Distance ( μm)**

**Response**

**ΔResponse**

**Pearson r**

**p-value (%)**

0 hours

0-50

0.236

0.031

0.28

2.49

•

50-100

0.279

0.018

0.37

3.5 × 10^{-3}

••••

100-150

0.322

0.0231

0.6

1.0 × 10^{-5}

••••

150-200

0.155

0.024

0.22

6.74

2 hours

0-50

0.488

0.04

0.48

0.17

••

50-100

0.296

0.025

0.55

1.1 × 10^{-3}

••••

100-150

0.214

0.032

0.35

1.06

•

150-200

0.097

0.026

0.29

2.52

•

4 hours

0-50

0.716

0.058

0.65

0.01

•••

50-100

0.42

0.025

0.56

3.3 × 10^{-4}

••••

100-150

0.376

0.028

0.55

2.1 × 10^{-3}

••••

150-200

0.251

0.035

0.46

0.21

••

heat-treated

0-50

-0.071

0.036

-0.091

54.97

50-100

-0.064

0.017

-0.08

32.52

100-150

0.051

0.015

0.112

9.16

150-200

-5.7 × 10^{-3}

0.015

0.027

68.39

Responses reported are the unitless slope _{tip }and sin _{mp}. The column ΔResponse is the standard error of this slope. The significance levels reported are for the Pearson

These correlations provide an estimate of the range of response that is consistent with our previous _{
closer
}/_{
farther
}and Δ_{
tip
}). The data evidence two trends for this response: it increases with longer incubation times and decreases at farther distances from the micropyle (Fig.

Although pollen tubes are known to turn in response to changes in their internal tip-focused cytoplasmic calcium gradient

The turns pollen tubes make are well-described by a model where ovules continuously release an attractant and pollen tubes respond to this attractant by following its gradient

Our experimental results show that pollen tubes change their direction of growth in a manner consistent with responding to a change in concentration across their tip, and that this response increases both with longer incubation times and as pollen tubes grow closer to the micropyle. To test and refine the mechanisms suggested by these data, we developed a mathematical model that encompasses both the release of an attractant by the ovules and the subsequent response that pollen tubes have to the attractant. Existing models of pollen tube behavior have focused on the physical processes that underlie tube growth, where cell shape, turgor pressure, internal ion gradients, and vesicle trafficking are essential considerations. Most models describe general tip growth in plants and fungi

We modeled how pollen tubes change their direction of growth by splitting each turn into a directed and a random component (Fig.

Model of pollen tube growth

**Model of pollen tube growth**. (A) Conceptual depiction of the directed and random components of turning. The directed component (black arrow) is calculated based on the gradient of the attractant. The random component is a random angle added to this. The gray shaded regions depict one standard deviation of the Gaussian distribution for the random angle. (B) Dynamics of a model of the gradient. The model gives a theoretical concentration of the attractant (Eq. 3 in Methods), and the gradient is derived from this concentration. Here the magnitude of the gradient from a single ovule, oriented toward the ovule micropyle is shown. Top: Depiction of the model for the attractant gradient as a function of distance from the micropyle. The different curves (top to bottom) are for the gradient after the source has released the attractant for 4.5 hours, 2.5 hours, and 0.5 hours. Bottom: Depiction of the model for the attractant gradient as a function of time on the medium. The different curves (top to bottom) are for distances of 0 μm, 50 μm , 100 μm , and 150 μm from the micropyle.

An exact model of spatial sensing would depend on both the distribution of receptors in these patches (or across the whole tip), the kinetics of the receptor-ligand interaction, and the nature of the intracellular response that ultimately results in the pollen tube turning. The distance and time scales in our experiment are large enough that we can assume receptors operate close to steady-state. We simplify the remaining considerations by assuming that the change in concentration across the tip (Δ

where Δ

To relate this model to the data in our experiments, we introduced a model for a relative concentration profile of the attractant (Fig.

Modeling the difference in concentration across the tip of the pollen tube requires relating how the concentration at the tip changes as the position of the tip changes. As discussed in Section 2.2, we expect Δ_{
tip
}sin _{
mp
}(Fig. _{
tip
}decreases with distance (the turning response increases closer to the micropyle) and increases with time (the turning response increases with longer incubation times).

When we combine the model for the direction of pollen tube growth and the attractant gradient, there are four parameters that describe the mean direction that the tubes turn in response to an attractant: the turning constant (_{
p
}), the attractant diffusion constant (_{0}). However, the parameters _{
p
}, and _{
p
}/_{0 }as fitting parameters (Table

Parameters for the turning model

**Parameter**

**Description (units)**

**Fit value**

**90% CI**

Proportional response (rad/conc min)

40.11

34.50-63.91

Diffusion constant (^{2}/min)

66.72

63.63-96.69

_{0}

Radial offset (

117.56

116.01-174.61

Validating the model

**Validating the model**. (A) Comparison of experimental results with the model. The responses are defined as in Fig. 3, where the response is the slope of the regression line between the turning angle _{tip }and sin _{mp}, which projects the gradient onto the tip of the pollen tube (see Fig. 3A). The different bars compare pollen tube responses observed in experiment, predicted from the model fit, and produced by simulations of the model. (B) Mean ⟨cos_{tip}⟩ plotted against _{tip}. The cosines of these angles are averaged for all points along the path separated by _{tip}⟩ as a function of _{0}) predicted from the linear fit in panel B, ^{2 }= 2

The fit yields a diffusion constant of 66.72 ^{2}/min, or 0.11 × 10^{-7 }cm^{2}/sec (Table ^{-7 }cm^{2}/sec in aqueous solution

Deviations from the mean direction of turning are consistent with how pollen tubes turn in the absence of ovules

Up to this point, our analysis has been used to understand the mean response of pollen tubes to the attractant, which is presumed to be released by ovules. We now turn to studying the substantial variation in response that pollen tubes exhibit _{
tip
}(_{
tip
}peaks sharply around _{
tip
}= 0, such that ⟨cos _{
tip
}⟩ ≈ 1 - ⟨_{
tip
}
^{2}⟩ and that the probability distribution can be described as a sharply-peaked Gaussian with variance ⟨_{
tip
}
^{2}⟩ = 2 (

Above, we assumed that the random component of growth is independent of the concentration of attractant. To test this idea, we ran simulations of our model that included both directed and random growth, with ovule locations and initial pollen tube locations and directions of growth taken directly from the corresponding experiments. We then treated these simulations as artificial time-lapse data and analyzed them in the same way that we analyzed our experimental data (see Methods). We found that the mean responses (directed component) in the simulations, as measured by the slope of the regression line between _{
tip
}and sin _{
mp
}, compared well to the data at different distances and for different incubation times (Fig. _{
tip
}predicted by the regression and the actual _{
tip
}. We compared the standard deviations of the populations of these residuals for both the simulations and the experiments (Table

Comparison of variations in responses in experiments and simulations.

**Distance**

**( μm)**

**Experiment**

**(radians)**

**Simulation**

**(radians)**

0 hours

0-50

0.747 ± 0.083

0.283 ± 0.006

50-100

0.550 ± 0.051

0.278 ± 0.003

100-150

0.324 ± 0.047

0.271 ± 0.003

150-200

0.335 ± 0.085

0.269 ± 0.003

2 hours

0-50

0.657 ± 0.078

0.306 ± 0.006

50-100

0.332 ± 0.029

0.291 ± 0.003

100-150

0.314 ± 0.036

0.281 ± 0.003

150-200

0.235 ± 0.023

0.268 ± 0.003

4 hours

0-50

0.602 ± 0.100

0.311 ± 0.007

50-100

0.420 ± 0.038

0.288 ± 0.004

100-150

0.383 ± 0.054

0.283 ± 0.003

150-200

0.264 ± 0.025

0.272 ± 0.003

Comparison of the circular standard deviations of turns in simulation and experiment. This summarizes the deviations from the mean turning response, which we treat as the random component of growth. This random component was calculated from the residual deviations between the mean response and the individual responses.

Incubation time influences the rate of growth near the micropyle

When we measured the persistence length of pollen tubes, we observed that the tubes began growing with an average rate of 2.76 ± 0.05 _{
farther
}frequencies at 0-50

Analysis of the growth rate near the micropyle

**Analysis of the growth rate near the micropyle**. (A) The average rate of growth depends on the distance to the micropyle. When pollen tubes grow within 50

In simulations, reducing the rate of growth increased the ability of pollen tubes to target ovules

To explore how this reduced growth rate would influence the guidance process, we added terms to our simulation to decrease the rate of growth with an increase in the gradient of the attractant (see Methods). These terms do not assume any particular molecular model for why the pollen tube slows, but their inclusion allowed us to include or exclude either a turning response (Eq. 1) or a slowing response (Eq. 4, in Methods). To assess the role that turning and slowing play in guidance, we calculated the fraction of tubes that were successfully able to target ovules in simulations for tubes that included or excluded these terms (Fig.

These results assumed the same turning response, which indicates that slower pollen tube growth near the micropyle can increase the ability of the tube to correctly target the micropyle without requiring the mechanisms that drive turning (the receptors at the tip) to increase their sensitivity. This effect can be understood geometrically by calculating how much a pollen tube must turn to prevent the angle _{
mp
}from increasing as it grows. Consider a pollen tube that grows at a rate _{
mp
}> 0°). If the pollen tube does not turn, over short times (_{
mp
}increases by _{
mp
}= (_{
mp
})_{
mp
}. For a turning response of _{
tip
}= _{
mp
})_{
mp
}is

By this reasoning, the angle _{
mp
}will only decrease (_{
mp
}< 0) when the response to the gradient

Discussion and conclusions

Previous semi-

Although the recent identification of pollen tube attractants in

Our model of the turning response did not assume a particular molecular mechanism for the sensing process. Some molecular features of pollen tube growth are likely to be important to include in future models as appropriate data become available. Pollen tubes have been observed to reorient in response to changes in the tip-focused cytoplasmic calcium gradient

Some features of pollen tube growth are unique among studied chemotropic responses. The eukaryotic cells commonly used in studies of chemotaxis,

Our results support a long-ranged chemotropic model for pollen tube guidance where pollen tubes respond to stable gradients maintained by ovules continuously releasing an attractant. Similar mechanisms have been proposed in genetic studies of the guidance process, but it is unknown how the attraction observed

The range of attraction _{
p
}/_{
p
}/

At the same time, a signal that provided funicular guidance need not have a very long range on the ovary placenta. Random motility has been shown to provide efficient search strategies in many organisms and would explain the variance in pollen tube growth seen in the ovary chamber

While it is unclear how the decrease in the rate of growth we observe near the micropyle

Methods

Plant growth and materials

As in

Semi-

Medium was modified slightly from

Stigmas, pollen, and ovules were derived from flowers selected at stage 14

Microscopy

Time-lapse images of GFP-labeled pollen were acquired using an Olympus Fluoview 1000 scanning confocal microscope. Positions of the ovules and stigma were determined using autofluorescence observed with a Cy5.5 filter.

Correcting stack alignment

The total fluorescence measured at 540 nm in each optical section was used to detect the surface of the medium. Each slice _{
j0}(Δ

Analysis of images

Pollen tube trajectories were constructed by using ImageJ image analysis software

Except for the _{
closer
}and _{
farther
}frequencies, we only included data from pollen tubes growing toward an ovule that was eventually, but not yet, penetrated by a pollen tube to ensure that our conclusions were based on data for guidance toward functional, unfertilized ovules. This restriction was not possible for the heat-treated control because the ovules were never penetrated in that case. The heat-treated control, in these cases, allowed a comparison of growth of pollen tubes between ovules that were capable of attracting the tubes and objects (heated-treated ovules) that were not. This provided a view of how random, or unguided, growth would appear in these measurements.

The angles Δ_{
mp
}, and _{
tip
}were calculated for turns in the plane perpendicular to the Z-axis of the confocal stacks, effectively projecting the confocal slices onto a single plane. Distances were confined to this plane to maintain consistency. Values of Δ**v**
_{
tip
}(**r**(**r**(**r**(**r**
_{
ov
}to form a vector **v**
_{
ov
}(**r**(**r**
_{
ov
}. We calculated the angle between these vectors to yield a value Δ

Values of _{
mp
}and _{
tip
}were calculated using three positions (at **v**
_{
cur
}(**r**(**r**(**v**
_{
new
}(**r**(**r**(**v**
_{
ov
}(**v**
_{
cur
}(t) and **v**
_{
new
}(t) was denoted _{
tip
}, and the angle between **v**
_{
cur
}(**v**
_{
ov
}(_{
mp
}.

Descriptive statistics of angular data

Normally the standard deviation of a sample provides a concise summary of the spread of a unimodal distribution. However, because Δ

Each angle Δ_{
i
}from a sample of angles is equivalent to a vector of length unity on a circle: **u**⟩:

The mean direction ⟨Δ**u**⟩ is expressed in polar coordinates:

In this expression ⟨Δ**u**⟩ and thus the mean direction of the is population of angles {Δ_{
i
}}. The length of ⟨**u**⟩ is _{0 }is chosen to correspond to the standard deviation of a normal distribution whose tails have been wrapped around a circle _{0}: values of σ_{0 }≈ 0 indicate a very narrow distribution, while values of σ_{0 }→ ∞ indicate an essentially uniform distribution of directions around the circle. To see this, consider **u**
_{
i
}will be almost identical, their sum will be a vector with length close to **u**⟩ will be close to _{0 }≈ 0. In the uniform distribution case, the vectors will be uniformly scattered so their directions will essentially cancle; the mean vector will be ⟨**u**⟩ ≈ 0, with length _{0 }will diverge (_{0 }→ ∞).

Standard errors, confidence intervals, and tests for statistical significance

Standard errors for _{
closer
}and _{
farther
}frequencies were calculated by treating each as an estimate of a Bernoulli trial probability, the standard error of which is ^{2 }testing _{
farther
}and _{
closer
}were initially tested with a 2 × 4 table of dichotomous outcomes: _{
farther
}at all distances and for distances of 50-100

Standard errors on the circular mean and circular standard deviation of the Δ_{0}[1]/_{0}[2]), where _{0}[1] and _{0}[2] were the resampled circular standard deviations, with 10,000 different permutations to test for significantly different circular standard deviations. Bootstrap calculations and permutation tests were performed in the R analysis package

Standard errors in the _{
tip
}angles were calculated by propagating the error in measuring the positions of each tip of the tube. Using the standard propagation of uncertainty, the standard error in the angle _{1 }and _{2 }is given by: (_{
θ
})^{2 }= (_{1}/_{1})^{2 }+ (_{2}/_{2})^{2}. Based on the size of the boxes used to track the tips, we assumed an isotropic standard error of 2 pixels (4

Standard error on the mean growth rate was estimated as

Model of directed turning

Our model for pollen tube response uses the well-studied Langevin equation

The first term describes turning that is proportional to the difference in the fraction of receptors at steady-state bound by the attractant at each side of the tip (Fig. _{
D
}(_{
D
}), Δ

Model for the ovule-secreted attractant

To propose a model for the concentration of the attractant, we proceeded similarly to previous models that have studied stable gradients _{
p
}) at the ovule micropyle. The concentration _{
tip
}=

Here _{1}(⋯) is the exponential integral, a well-characterized special function _{0 }is an offset we introduced to account for the distance that the attractant has to diffuse on the surface of the ovule, where the diffusion coefficient may be different, before entering the thin film of liquid pollen-growth medium that coats the top of the solid agar matrix. This offset also corrects for non-physical behavior near the origin, where a finite amount of attractant is deposited into an infinitesimal region, making the concentration there infinite

To determine how experimental errors would affect the model parameters, we used our Gaussian model for the error in tip positions (s. d. of 4

Model of random turning

To access short regions of growth, we took advantage of the fact that, on the length scales of the experiment, pollen tube growth is smooth with few sharp angles and fit the points in the time-lapse with a spline curve to study the growth at intervals as short as 5 _{
tip
}(_{
tip
}(**v**(**v**(_{
tip
}(**v**(**v**(**v**(_{
s
}= ⟨(cos_{
tip
}(_{
s
}, where the subscript

where the last approximation is valid for short amounts of growth relative to the persistence length (_{
tip
}⟩ against _{
tip
}(^{-4 }and _{
tip
}peaks sharply around _{
tip
}= 0, indicating that ⟨cos_{
tip
}⟩ ≈ 1 - ⟨_{
tip
}
^{2}⟩. and that the distribution of _{
tip
}can be described as a sharply-peaked Gussian with mean 0 and variance

Fitting parameters

Given the parameters _{
p
}, _{0}, our model calculates the mean direction a tube should turn from (a) the location of a pollen tube tip, (b) the direction the tip is growing, (c) the time the ovules have had to release a guidance cue, and (d) the location of the ovules. Our experimental pollen tube trajectories contained both this information and the angle the tube actually turned. We used a ^{2 }metric to evaluate how well a set of parameters described the experimental data:

Here _{
tip
}includes both the error in measuring _{
tip
}and the fluctuations predicted by the _{
i
}is the predicted mean angle for turning:

where the subscript _{
mp
}(**r**
_{
i
}and occurs at time _{
i
}, and Δ**r**
_{
mp
}(_{
i
}) is the change in concentration across the tip of the pollen tube, which depends on the same quantities. Here the width of the pollen tube, Δ

Fits obtained using the Levenberg-Marquardt algorithm to minimize ^{2 }
**r**, _{
mp
}, _{
p
}/_{
p
}, and _{
p
}/_{0 }the mean response _{
i
}is then

where we write _{
p
}/_{0}. To understand the resulting fits, we assessed the turning response of the model and compared this response at different distances with the experimental responses (Fig.

Model for pollen tube growth rates

To model how the growth rate decreased near the micropyle, we assumed that the pollen tubes were responding to higher gradients of the attractant. A simple way to model this is to assume that a pollen tube periodically adjusts its rate of growth based on the difference in the concentration it perceives across its tip: _{
new
}= _{
min
}+ (_{
old
}- _{
min
})/(_{
v
}Δ_{
v
}modulates the response to the change in concentrations. At low values of Δ_{
new
}≈ _{
old
}, while at high values, _{
new
}approaches _{
min
}. This formulation is consistent with our general observation that once the rate of growth of a pollen tube had slowed, it never substantially increased. Our model is essentially a continuously-sampled formulation of _{
new
}:

where _{
min
}= 0.5 _{
v
}and _{
v
}= 533.39 (1/dimensionless concentration units) and

Simulation protocol

Each simulation has a set of virtual pollen tubes, each of which has an index **r**
_{
j
}, as well as a rate and a direction of growth (the magnitude and direction of vector **v**
_{
j
}). In addition, each simulation also has a list of ovule micropyle locations. We implement the model (Eqs. 1 and 4) by choosing discrete time-steps of length Δ

1. If the virtual pollen tube has been previously "captured" by coming within a short distance (10

2. The virtual tip is advanced based on the previous direction: **r**
_{
j
}(**r**
_{
j
}(**v**
_{
j
}(

3. The concentration and gradient at position **r**
_{
j
}(_{
p
}/

4. We pick a random number _{
j
}from a Gaussian distribution with mean ^{2 }= 2_{
j
}(_{
mp
}is the angle between the direction **v**
_{
j
}(

5. In simulations where the rate of grow changes (denoted S+ in the text), the new rate of growth _{
j
}(_{
j
}(**v**
_{
j
}(

6. **v**
_{
j
}(**v**
_{
j
}(_{
j
}(

7. If the new position of the virtual pollen tube, **r**
_{
j
}(

These steps are continuously iterated until the simulation ends.

In addition to this algorithm, each simulation requires the location of the virtual ovules and a set of initial positions and directions for the virtual pollen tubes. To enable direct comparison between our simulations and the experimental data, we used the same micropyle locations, initial pollen tube locations, and incubation time as an experimental replicate. Each group of incubation times simulated (0-hr, 2-hr, and 4-hr) used the same replicates as in the experiments. To set the initial pollen tube directions, we used a vector between the first and second experimental positions for each tube, scaled by the time interval between the measurements (20 min). Our simulations were unconstrained by the requirements of image analysis, which meant that we could run an unlimited number of pollen tubes in each virtual replicate, and we chose to use 500 tubes in each replicate to increase the statistics of each run. Each experimental replicate had 20-40 tubes, which gave us 20-40 initial conditions (positions and directions). We randomly, and uniformly, chose one of these initial conditions for each of our virtual pollen tubes, essentially treating the experimental set of initial conditions as a bootstrap distribution. Because of the random variations in the turning angle (the random number in step 4), two virtual tubes that start with the same initial condition will ultimately have distinct paths. This mirrors the experimental behavior where many of the tubes initially emerged from the transmitting tract in a tightly packed formation, growing in the same direction, but then the tubes would grow randomly on the medium, spreading out to a fan-like distribution.

Authors' contributions

SFS performed the experiments and analyzed the data described in this study. SFS, MJR, DP, and ARD conceived, designed, coordinated this study and drafted the manuscript. PB and MT contributed simulation and analysis tools to this study. All authors read and approved the final manuscript.

Acknowledgements

Funding for this project was provided by the Burroughs-Wellcome Fund Interfaces ID 1001774 and DOE Grant DE-FG02-96ER20240. We would like to thank Ravi Palanivelu for helpful discussions that led to the initiation of this project, and Karen Reddy for her technical expertise in microscopy. We would also like to thank Caroline Taylor, Maria Krisch, Beth Bray, and Mark Johnson for their helpful comments on the manuscript.