### Plant growth and materials

As in [7], pollen was derived from LAT52:GFP transgenic lines in a Columbia background. Stigmas, styles, and ovules were from the *A. thaliana* male sterile mutant, *ms1* (CS75, Landsberg background). Seeds were sown in soil and stratified at 4°C for 2 days, and plants were grown under fluorescent light (100 *μ*E) for 16 or 24 hrs/day at 40% humidity.

### Semi-*in vitro* pollen guidance assay

Medium was modified slightly from [7] by embedding 1 *μ*m FluoroSphere fluorescent beads that emitted at 540 nm (Invitrogen). Beads were used to correct for drift along the Z-axis in the confocal stacks taken of the system. The presence of the beads did not affect the growth or response of the pollen tubes.

Stigmas, pollen, and ovules were derived from flowers selected at stage 14 [62]. Stigmas were cut at the junction between the style and the ovary using surgical scissors (World Precision Instruments, Sarasota, USA), and were placed horizontally on the pollen growth medium. Stigma were pollinated on the medium with 30-60 pollen grains. Pollen tubes began to emerge two hours after pollination. Ovules were excised from the ovary by first removing the ovary wall with the tip of a 27 gauge needle and then excising the ovules by cutting at the base of the funiculus using a Minutien pin (Fine Science Tools, Foster City, CA, USA). The excised ovules were removed from the ovary and deposited on the surface of the medium, where they were then arranged around the cut stigmas. A 001 insect pin mounted in a pin vice (Fine Science Tools) was used for removal and subsequent manipulation of the ovules. The timing of pollinating the stigmas and placing the ovules was varied according to the desired ovule incubation time (Table 1). For 0-hour incubation times, the stigma was pollinated and two hours later the ovules were placed. For 2-hour incubation times, the ovules were placed, and the stigmas were then immediately pollinated. For 4-hour incubation times, the stigmas were pollinated two hours after the ovules were placed.

### 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.

### Analysis of images

Pollen tube trajectories were constructed by using ImageJ image analysis software http://rsb.info.nih.gov/ij/download.html. Kalman filtering, as implemented by the Kalman Stack Filter plugin to ImageJ by Chris Mauer, was applied to the stacks before image analysis http://rsb.info.nih.gov/ij/plugins/kalman.html. The tips of the pollen tubes were identified manually. The micropyle of each functional ovule was located by the point where a pollen tube had entered the micropyle, penetrating the ovule. This penetration was assessed with two conditions: pollen tubes had to both reach the micropyle, and subsequent growth had to occur within the focal planes of the ovule autofluorescence. The micropyles of heat-treated ovules were taken to be at the location of the cleft where the funiculus joins the outer integument of the ovule.

Except for the *f*
_{
closer
}and *f*
_{
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*θ*, *θ*
_{
mp
}, and *θ*
_{
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*θ* were calculated as follows (see Fig. 2C). In each single tube, we subtracted the tip positions at each pair of adjacent time-points *t* and *t*+Δ*t* to give a vector of the growth direction **v**
_{
tip
}(*t*) = **r**(*t* + Δ*t*) - **r**(*t*). We the subtracted the position **r**(*t*) from the position of the closest micropyle **r**
_{
ov
}to form a vector **v**
_{
ov
}(*t*) = **r**(*t*) - **r**
_{
ov
}. We calculated the angle between these vectors to yield a value Δ*θ*(*t*) for each time-point *t* in a single tube path.

Values of *θ*
_{
mp
}and *θ*
_{
tip
}were calculated using three positions (at *t* - Δ*t*, *t*, and *t* + Δ*t*) as follows. We calculated the vector of the current growth direction: **v**
_{
cur
}(*t*) = **r**(*t*) - **r**(*t* - Δ*t*). The new growth direction was the calculated similarly: **v**
_{
new
}(*t*) = **r**(*t* + Δ*t*) - **r**(*t*). The direction to the micropyle **v**
_{
ov
}(*t*) was calculated as in Δ*θ*. For each point, the angle between **v**
_{
cur
}(t) and **v**
_{
new
}(t) was denoted *θ*
_{
tip
}, and the angle between **v**
_{
cur
}(*t*) and **v**
_{
ov
}(*t*) was denoted *θ*
_{
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*θ* is an angle, we cannot use linear statistics to describe it. To understand this issue, consider a sample of two angles, 1° and 359°. The linear mean of these angles is (1°+359°)/2 = 180° and the linear standard deviation is 253.1°. However the actual mean direction of these angles is 0°, and the correct circular standard deviation is 1.4°. Because angles are periodic, 359° corresponds to -1°. A correct statistical description of a sample of angles is given by circular statistics [26, 27], in which angles are mapped to unit vectors on a circle. This transformation allows the correct calculation of the mean direction and gives a natural circular equivalent to the linear standard deviation; we describe it briefly.

Each angle Δ

*θ*
_{
i
}from a sample of angles is equivalent to a vector of length unity on a circle:

. The direction ⟨Δ

*θ*⟩ of the angles is found by calculating the mean vector ⟨

**u**⟩:

The mean direction ⟨Δ

*θ*⟩ is apparent when ⟨

**u**⟩ is expressed in polar coordinates:

In this expression ⟨Δ*θ*⟩ is the direction of the mean vector ⟨**u**⟩ and thus the mean direction of the is population of angles {Δ*θ*
_{
i
}}. The length of ⟨**u**⟩ is *R*, which gives a measure of the spread of the vectors around the circle. The sample circular standard deviation is related to *R* by
. This form for σ_{0} is chosen to correspond to the standard deviation of a normal distribution whose tails have been wrapped around a circle [26], and our intuition for Gaussian distributions can be similarly applied to σ_{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 *N* angles chosen from a distribution with a very narrow spread around Δ*θ* = 0, and *M* angles chosen from a distribution that is uniform around the circle. In the narrow distribution, the unit vectors **u**
_{
i
}will be almost identical, their sum will be a vector with length close to *N*, and the length of the mean venctor ⟨**u**⟩ will be close to *R* ≈ 1, so *σ*
_{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 *R* ≈ 0, and *σ*
_{0} will diverge (*σ*
_{0} → ∞).

### Standard errors, confidence intervals, and tests for statistical significance

Standard errors for *f*
_{
closer
}and *f*
_{
farther
}frequencies were calculated by treating each as an estimate of a Bernoulli trial probability, the standard error of which is
[63]. Significant differences between these frequencies were determined by *χ*
^{2} testing [63], implemented in the R analysis package [64]. Differences between the frequencies *f*
_{
farther
}and *f*
_{
closer
}were initially tested with a 2 × 4 table of dichotomous outcomes: *p* < 0.01 for *f*
_{
farther
}at all distances and for distances of 50-100 *μ*m. Differences in
were initially tested with a 2 × 3 table: *p* < 0.01 for distances of 0-100 *μ*m. Significance of the pairwise comparisons was tested with a 2 × 2 table.

Standard errors on the circular mean and circular standard deviation of the Δ*θ* angle were calculated using a bootstrap method with 1000 resamples for each statistic [65]. We used a one-sided computational permutation test of the statistic log(*σ*
_{0}[1]/*σ*
_{0}[2]), where *σ*
_{0}[1] and *σ*
_{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 [64–67]. The confidence interval in the linear model describing persistence was calculated from 10,000 samples generated using the Monte Carlo method included in the program pro Fit [68].

Standard errors in the *θ*
_{
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 *θ* between two lines of length *l*
_{1} and *l*
_{2} is given by: (*σ*
_{
θ
})^{2} = (*σ*
_{1}/*l*
_{1})^{2} + (*σ*
_{2}/*l*
_{2})^{2}. Based on the size of the boxes used to track the tips, we assumed an isotropic standard error of 2 pixels (4 *μ*m) for the position of the tips of the pollen tubes.

Standard error on the mean growth rate was estimated as
, where *SD* is the maximum likelihood estimate of the standard deviation. The significance of pairwise comparisons of growth rates was determined using Tukey's honest significant difference with a 95% family-wide confidence interval.

### Model of directed turning

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

69] which separates turning into directed and random components:

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. 3A), and the second term adds a random variation to the receptor-mediated response. Here *κ* is the proportionality constant for turning, *c* is the concentration of attractant in units of units of *K*
_{
D
}(*C* = *c K*
_{
D
}), Δ*c* is the change in concentration across the tip of the pollen tube, *ξ*(*t*) is a random process that is uncorrelated in time and is Gaussian-distributed with unit variance, and *σ* is the magnitude of the noise. Initial fits of the model to our data revealed no saturation in the turning response, allowing us to simplify the first term. Our model for the receptor response then reduces to Eq. 1.

### 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 [

45–

47]. We described diffusion of the attractant on the medium with Fick's law and its release from the ovule with a constant source have radial (rate

*k*
_{
p
}) at the ovule micropyle. The concentration

*c*(

*r*,

*t*) and gradient

*G*
_{
tip
}=

*dc*(

*r*,

*t*)/

*dr* have radial symmetry, and their solution as a function of the distance from the origin

*r* is

Here *D* is the diffusion constant of the attractant and *E*
_{1}(⋯) is the exponential integral, a well-characterized special function [70, 71]. The parameter *r*
_{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 [47].

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 *μ*m) to generated 10,000 synthetic data sets. To estimate confidence intervals for the model parameters, we fit theses data sets. Table 3 reports the 90% range of values for these fits.

### 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

*μ*m of growth. The change in direction of a pollen tube was found by finding the angle between the direction of growth at some distance along the tube

*s* and a new direction of growth after the tube had grown a distance

*δs* (Fig.

5B). We write this as the turning angle

*θ*
_{
tip
}(

*s*,

*s* +

*δs*). Its cosine, cos

*θ*
_{
tip
}(

*s*,

*s* +

*δs*), measures the correlation between the vector for the direction of growth at

*s* and the vector at

*s* +

*δs*:

**v**(

*s*)·

**v**(

*s* +

*δs*) = cos

*θ*
_{
tip
}(

*s*,

*s* +

*δs*), where

**v**(

*s*) has been normalized to have unit length. Averaging this quantity over all lengths of

*δs* provides a measure of the average amount of the original direction retained in the new direction: ⟨

**v**(

*s*)·

**v**(

*s* +

*δs*)⟩

_{
s
}= ⟨(cos

*θ*
_{
tip
}(

*s*,

*s* +

*δs*)⟩

_{
s
}, where the subscript

*s* indicates an average over all possible lengths

*δs*. This correlation function often has an exponential form:

where the last approximation is valid for short amounts of growth relative to the persistence length (*δs* ≪ *L*). Graphically plotting ⟨cos*θ*
_{
tip
}⟩ against *δs* revealed a roughly linear relationship, with an intercept close to unity and a slope just below zero, consistent with a long persistence length (Fig. 5B). Specifically, we fit this relationship with the linear model ⟨cos*θ*
_{
tip
}(*δs*)⟩ = (1 + *b*) - *δs*/*L*, with *b* = -7.0 × 10^{-4} and *L* = 1074.02 *μ*m. The parameter *b* is the deviation from the intercept at unity and was not statistically different from *b* = 0. The parameter *L* is the persistence length, and was found to fall in range 1042.70-1108.68 *μ*m with 99.9% confidence. Although the data showed non-random oscillations around the linear fit, the deviations were small (the two largest deviations were a 5% error at *δs* = 115 *μ*m and 6% error at *δs* = 385 *μ*m). The long persistence length *L* indicates that the probability of making a turn *θ*
_{
tip
}peaks sharply around *θ*
_{
tip
}= 0, indicating that ⟨cos*θ*
_{
tip
}⟩ ≈ 1 - ⟨*θ*
_{
tip
}
^{2}⟩. and that the distribution of *θ*
_{
tip
}can be described as a sharply-peaked Gussian with mean 0 and variance
. This allows us to set the parameter *σ* which scales the random component of a turn to
.

### Fitting parameters

Given the parameters

*κ*,

*k*
_{
p
},

*D*, and

*r*
_{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

*χ*
^{2} metric to evaluate how well a set of parameters described the experimental data:

Here

*σ*
_{
tip
}includes both the error in measuring

*θ*
_{
tip
}and the fluctuations predicted by the

*σ* parameter of the model. The term

*θ*
_{
i
}is the predicted mean angle for turning:

where the subscript *i* denotes each separate direction to the micropyle *θ*
_{
mp
}(*i*), which has position **r**
_{
i
}and occurs at time *t*
_{
i
}, and Δ*c*(**r**
*i*, *θ*
_{
mp
}(*i*), *t*
_{
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*L*, is absorbed into the constant *κ* with no loss of generality.

Fits obtained using the Levenberg-Marquardt algorithm to minimize

*χ*
^{2} [

71] often became stuck in local minima. We found that a Powell's level set method [

71], implemented in Scientific Python [

72,

73], proved much more robust. When fitting, the adjustable parameters should be made as independent as possible. The term Δ

*c*(

**r**,

*θ*
_{
mp
},

*t*) has a prefactor of

*k*
_{
p
}/

*D*, which makes the parameters

*κ*,

*k*
_{
p
}, and

*D* highly covariant. We removed this dependence by introducing a combined parameter

*κ*' =

*κ*(

*k*
_{
p
}/

*D*) and fitting

*κ*' and the diffusion constant

*D* as two independent parameters (

*D* can be left as a separate parameter because of its appearance in the exponential in Eq. 3). Written in terms of the parameters

*κ*',

*D*, and

*r*
_{0} the mean response

*θ*
_{
i
}is then

where we write
to emphasize that we removed the *k*
_{
p
}/*D* prefactor, but that these terms are still parameterized by *D* and *r*
_{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. 5A).

### 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:

*v*
_{
new
}=

*v*
_{
min
}+ (

*v*
_{
old
}-

*v*
_{
min
})/(

*k*
_{
v
}Δ

*c* + 1), where

*k*
_{
v
}modulates the response to the change in concentrations. At low values of Δ

*c*,

*v*
_{
new
}≈

*v*
_{
old
}, while at high values,

*v*
_{
new
}approaches

*v*
_{
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

*v*
_{
new
}:

where *τ* is the timescale for slowing growth in response to a gradient. To fit this model, we used the rate of growth between each time point with *t* = 20 min. We noticed that tubes would grow as slowly as 0.5 *μ*m/min when near the micropyle and accordingly set *v*
_{
min
}= 0.5 *μ*m/min. The parameters *k*
_{
v
}and *τ* were fit with the robust fitting method provided by the pro Fit analysis program [68] to obtain values *k*
_{
v
}= 533.39 (1/dimensionless concentration units) and *τ* = 19.20 min. The model showed good agreement with the average rate of growth until very close to the micropyle (*R* ~ 10 *μ*m), where it predicted a higher average rate of growth than observed.

### Simulation protocol

Each simulation has a set of virtual pollen tubes, each of which has an index

*j*, a current position

**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

*t* = 0.1 min and using the model equations to evolve the position and direction of growth of each virtual pollen tube tip. Specifically, at each step in time Δ

*t*, the simulation iterates through the list of pollen tubes and does the following:

- 1.
If the virtual pollen tube has been previously "captured" by coming within a short distance (10 *μ*m) of the virtual micropyle, or growing more than 800 *μ*m from the center of the simulation, then the simulation ignores it.

- 2.
The virtual tip is advanced based on the previous direction: **r**
_{
j
}(*t* + Δ*t*) = **r**
_{
j
}(*t*) + **v**
_{
j
}(*t*)Δ*t*

- 3.

- 4.

- 5.
In simulations where the rate of grow changes (denoted S+ in the text), the new rate of growth

*v*
_{
j
}(

*t* + Δ

*t*) is calculated according to Eq. 4, where the old rate of growth is of growth

*v*
_{
j
}(

*t*) the magnitude of

**v**
_{
j
}(

*t*):

- 6.
**v**_{
j
}(*t* + Δ*t*) is generated by rotating **v**_{
j
}(*t*) by *δθ*_{
j
}(*t* + Δ*t*) and rescaling the vector to have magnitude *v*(*t*+Δ*t*).

- 7.
If the new position of the virtual pollen tube, **r**
_{
j
}(*t* + Δ*t*), is within 10 *μ*m of the ovule icropyle or has grown more than 800 *μ*m from the center of the simulation, then the virtual tube is marked as "captured," and is ignored in subsequent iterations.

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.