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Detection of a Gravitropism Phenotype in glutamate receptor-like 3.3 Mutants of
Arabidopsis thaliana Using Machine Vision and Computation
Nathan D. Miller*#, Tessa L. Durham Brooks*1#, Amir H. Assadi§, Edgar P. Spalding*
*Department of Botany §Department of Mathematics
University of Wisconsin University of Wisconsin
Madison, WI 53706 Madison, WI 53706
Gene ID and Mutant stocks: AT1G42540, Salk_040458, and Salk_066009
1 Present address: Department of Biology
Doane College
Crete, NE 68333
#These authors contributed equally to this work.
Genetics: Published Articles Ahead of Print, published on July 20, 2010 as 10.1534/genetics.110.118711
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Running title: Machine vision phenotype detection
Key words: Gravitropism, phenotype detection, machine vision, Arabidopsis,
glutamate receptor
Author for correspondence: Edgar P. Spalding
University o f Wisconsin
Department of Botany
430 Lincoln Drive
Madison, WI 53706
Fax: 608-262-7509
Tel: 608-265-5294
Email: [email protected]
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ABSTRACT
Gene disruption frequently produces no phenotype in the model plant Arabidopsis
thaliana, complicating studies of gene function. Functional redundancy between gene
family members is one common explanation but inadequate detection methods could
also be responsible. Here, newly developed methods for automated capture and
processing of times series of images, followed by computational analysis employing
modified linear discriminant analysis (LDA) and wavelet-based differentiation were
employed in a study of mutants lacking the Glutamate Receptor-Like 3.3 gene. Root
gravitropism was selected as the process to study with high spatiotemporal resolution
because the ligand-gated Ca2+-permeable channel encoded by GLR3.3 may contribute
to the ion fluxes associated with gravity signal transduction in roots. Time series of root
tip angles were collected from wild type and two different glr3.3 mutants across a grid of
seed-size and seedling-age conditions previously found to be important to gravitropism.
Statistical tests of average responses detected no significant difference between
populations, but LDA separated both mutant alleles from the wild type. After projecting
the data onto LDA solution vectors, glr3.3 mutants displayed greater population
variance than the wild type in all four conditions. In three conditions the projection
means also differed significantly between mutant and wild type. Wavelet analysis of the
raw response curves showed that the LDA-detected phenotypes related to an early
deceleration and subsequent slower-bending phase in glr3.3 mutants. These
statistically significant, heritable, computation-based phenotypes generated insight into
functions of GLR3.3 in gravitropism. The methods could be generally applicable to the
study of phenotypes and therefore gene function.
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INTRODUCTION
A major objective of research on the model plant Arabidopsis thaliana is to determine
functions for each of its approximately 25,000 genes. An extensive, sequence-indexed
library of T-DNA insertion mutants has resulted in reverse genetics becoming a routine
approach toward this goal (Alonso and Ecker 2006). This approach is particularly
effective when the mutation results in an observable phenotype that gives a clue about
the disrupted gene’s function (Kuromori et al. 2006). Unfortunately, the large majority of
gene disruptions in Arabidopsis produce no readily observable phenotype (Bouché and
Bouchez 2001; Kuromori et al. 2006). To date, functional descriptions for only
approximately 10% of the Arabidopsis genes have been experimentally determined.
Reverse-genetic approaches in other organisms, such as C. elegans and Drosophila
have yielded similar results (Fraser et al. 2000). One possible explanation for the
infrequency of phenotypes is functional redundancy, especially when the gene is a
member of a large family. Or, the phenotype may be conditional, manifesting itself only
in a particular environment or developmental context that was not examined. Lastly, the
methodologies employed to search for a phenotype may not match well with the missing
gene’s function or scale of contribution. Detecting the effect of a mutation in only one of
the organism’s approximately 104 genes may require a specialized technique. In this
regard, high resolution measurements of growth over time hold much promise (van der
Weele et al. 2003; Beemster et al. 1998; Chavarría-Krauser 2006; Miller et al. 2007;
Reddy and Roy-Chowdhury 2009; Spalding 2009).
One of the surprises to come from the first plant genome sequencing effort was
the presence of 20 Arabidopsis genes homologous with those encoding mammalian
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ionotropic glutamate receptors (Lam et al. 1998; Lacombe et al. 2001). Because the
animal molecules were known almost exclusively as tetrameric ion channels mediating
synaptic transmission in the central nervous system (Mayer and Armstrong 2004), their
presence in plants attracted considerable attention. The first studies explored the
structure and evolution of the plant gene family (Turano et al. 2001; Chiu et al. 2002).
Subsequent studies employing antisense methods to reduce expression and
constitutive promoters to overexpress GLR family members indicated possible roles in
coordinating carbon and nitrogen metabolism (Kang and Turano 2003), abscisic acid
biosynthesis and signaling (Kang et al. 2004), Ca2+ and Na+ homeostasis (Kim et al.
2001), Ca2+ and fungal disease progression (Kang et al. 2006), and Ca2+-mediated
stomatal closure (Cho et al. 2009). Transcription of multiple family members in the same
cell type made heteromeric channels seem probable in planta (Roy et al. 2008). While
each study provides clues, a consistent theme has not emerged. A robust mutant
phenotype could give very useful direction to further experimentation but none has been
reported.
The Arabidopsis GLR genes are different enough from the animal
neurotransmitter-gated channels in key regions, such as the putative ion-conducting
pore and extracellular amino-terminal domains, that equivalent molecular function
cannot be assumed (Davenport 2002). But the demonstration that wild-type plants
respond to glutamate with a strong membrane depolarization and fast transient rise in
cytoplasmic Ca2+ concentration made ligand-gated channel activity for the plant GLR
molecules a viable hypothesis (Dennison and Spalding 2000). The hypothesis was
strongly supported when these ionic events were found to be blocked by mutations in
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the GLR3.3 and GLR3.4 genes (Qi et al. 2006; Stephens et al. 2008). In spite of these
strong ionic phenotypes, growth or development defects that typically guide hypotheses
about gene function could not be found. Either such phenotypes do not exist in glr3.3
mutants, or some nonstandard methods for finding them were necessary. Here, a
highly automated process for quantifying dynamic root growth and behavior involving
image processing and mathematical analysis was employed to search for a root growth
behavior phenotype (Miller et al. 2007; Durham Brooks et al. 2010). The results
demonstrate a function for GLR3.3 in root gravitropism and provide an example of how
a single-gene phenotype can be isolated by applying appropriate technology.
MATERIALS AND METHODS
Plant material
Arabidopsis thaliana (Columbia ecotype) seeds were sieved with grading sizes of
250µm2, 280 µm2, 300 µm2, and 355 µm2. Seeds between 250 µm2 – 280 µm2 were
classified as small and those from 300 µm2 – 355 µm2 were classified as large. Sieved
seeds were surface sterilized with 70% ethanol, 2% Triton X-100 and were planted on a
1% agar medium containing 1 mm KCl, 1 mm CaCl2, 5 mm 2-[N-morpholino]-
ethanesulfonic acid and pH adjusted to 5.7 with 1,3-
bis[tris(hydroxymethyl)methylamino]propane. After stratification at 4°C for 3–7 days, the
seeds were germinated on a vertically oriented plate and grown for 2-4 days under
50 μmol m−2 s−1 white light.
Mutant genotyping
Seeds of plant lines containing T-DNA insertions in GLR3.3 (At1g42540) were obtained
from the Salk Institute (http://signal.salk.edu/cgi-bin/tdnaexpress). The lines used were
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Salk_040458 (glr3.3-1, second exon insertion) and Salk_066009 (glr3.3-2, first intron
insertion). Homozygous individuals were genotyped using the method described
previously (Qi et al., 2006). The PCR products from amplification with the left border
primer were sequenced to verify the position of the insertion.
Imaging
Petri plates containing seedlings were mounted vertically and transverse to the optical
axis of one of seven CCD cameras (Marlin F146B, Allied Vision Technologies (AVT),
Newburyport, MA, USA, www.alliedvisiontec.com) outfitted with a close-focus zoom lens
(NT59-816; Edmund Optics, http://www.edmundoptics.com). An infrared backlight
(NT55-819, Edmund Optics) having a peak output at 880 nm, was positioned behind
each Petri plate for back illumination. Resulting images were 1392 × 1040 pixels at 8-
bit pixel depth, with a maximum resolution of approximately 5 μm per pixel. Only one
root per plate was analyzed, even if there were two or three present. An x,y,z
positioning device on the plate holder was used to pose the selected root in the center
of the frame. To initiate the experiment, the plate was rotated until the tip of the root was
horizontal as best judged by eye, i.e. within a degree or two of the camera’s horizon.
File-acquisition rate and storage of the images in tag image file format (TIFF) was
controlled by AVT software. Each camera acquired images of a seedling root every 2
min for 10 h beginning when the seedling was rotated to induce gravitropic root
bending. A total of 255 such ‘movies’ were acquired for the studies presented here. All
the components required for an imaging apparatus and a step-by-step assembly guide
may be found at http://phytomorph.wisc.edu/hardware/fixed-cameras.php.
Image analysis
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Using the image processing methods detailed in the Supplemental Material section
(http://botany.wisc.edu/spalding/PlantJournal2007/Supplemental_Material.htm) of Miller
et al. (2007), the midline was extracted from each root image. Tip angle was calculated
by first performing principal components analysis on a 5-pixel region of the midline near
the root tip. The tip angle was the angle formed between the first principal component
and the camera’s horizon. Growth rate was calculated as the differential of the midline
length over time. Growth rates between mutant and wild type roots were not different to
a statistically significant degree and were not used in the analyses presented here.
Linear discriminant analysis and its optimization
To find the projection of the data best satisfying each objective function, a minmax
optimizer in the optimization toolkit of the MATLAB scientific programming language
(Mathworks, www.mathworks.com/products) was applied to the entire population of tip
angle versus time points for 300 iterations. These 300 results were filtered to find the
solution of this population producing the smallest p-values between the mutant and wild-
type. To determine if the projection resulted in significant separation, tests of
significance were performed. A two-sample t-test was used to calculate significance of
the solution to Eq. 1 optimization. During the iterative search for variance separation
(Eq. 2 and Eq. 3), an F-test determined the significance level of each result. Statistical
significance of the final result was determined with a Brown-Forsythe test.
Wavelet analysis
Wavelet analysis was performed on each individual tip angle response using the first or
second order derivative of the Gaussian distribution as the transforming function with
window sizes from 1 to 20. To determine the significance of the wavelet-transformed
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data, t-tests were run between each mutant allele and the wild-type populations for each
scale at each condition. Regions of the response in which the mutant data differed from
the wild type were considered significant when p was less than 0.05 at any wavelet
scale in both alleles.
Fitting of wavelets to the LDA solution vector was performed as follows. The four
first order Gaussian derivatives that best correlated with the solution vector were
determined using a watershed algorithm. Then, pairwise combinations of these four
wavelets were least-squares fit to the solution vector. The pair with the best fit was
identified then summed to create the wavelet fit of the LDA solution vector.
RESULTS
A bank of seven CCD cameras each equipped with a close-focus zoom lens formed the
front end of a computer-controlled image acquisition and analysis platform that was
used in a previous study to investigate the plasticity of wild-type root gravitropism
(Durham Brooks et al. 2010). The size of the seed from which the seedling emerged
and post-germination age significantly affected response trajectory when measured with
high resolution at 2 min intervals over a 10 h period (Durham Brooks et al. 2010).
Therefore, seed size (small or large) and seedling age (2 d or 4d) created a 2 x 2
condition grid in which a gravity response phenotype was sought in two T-DNA insertion
(mutant) alleles of GLR3.3 (At1g42540). All image data acquired during this study are
available at http://phytomorph.wisc.edu/download.
Figure 1 shows the average time course of root tip angle after gravistimulation in
the wild type and two glr3.3 alleles in each of the four chosen conditions. As found in a
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recent characterization of the wild type (Col-0 ecotype), young seedlings responded
vigorously and transiently overshot the ultimate steady-state angle regardless of seed
size or genotype (A and B). Older seedlings more steadily approached the new vertical
upon re-orientation (C and D). Both glr3.3 alleles developed tip angle slightly differently
than the wild type in conditions B, C, and D (for example, note the initial response
rates), though t-tests indicated that the differences were not significant at any of the
time points (data not shown). However, a null result based on population averages
does not rule out an effect of the glr3.3 mutation on this root growth response.
Other methods for finding evidence of differences between populations of
measurements exist. Linear discriminant analysis (LDA), first devised by Fisher (1936)
to investigate a plant taxonomy question using sepal size, is one such method. A
method similar to Fisher’s original LDA for separating two groups was implemented to
determine if two groups (glr3.3-1 and glr3.3-2) could be separated similarly from a third
(the wild type). The input data were tip angle time points (301 per trial x n trials per
condition). They were treated as a high-dimension data cloud by recasting each time
course as a single point in 301-dimensional space. The next step was to design an
objective function that specified the hypothesis to test as follows. One objective
function sought a linear projection of the data that maximally separated the two mutant
population means from the wild-type mean while minimizing the standard deviations of
each, i.e.
min�
max���, �� min�
max � |����� ���|����� � ��� , |����� ���|����� � ��� � �. �. ��� 1 (Eq 1)
where:
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�� 1��
� ���
� is the expected value for the Jth group; i.e., wt, mut1, mut2
�
��� 1�� 1 ����
�
� ���� is the variance for the Jth group; i.e., wt, mut1, mut2
�
and
��
� is the ith trial for the Jth group
The second objective function sought a linear projection of the data that minimized the
variance of the wild-type population relative to the mutant populations, i.e.
min�
max���, �� min�
max � �������� , �������� � �. �. ��� 1 (Eq 2)
The third found a linear projection of the data that minimized the variance of the mutant
populations relative to the wild type population, i.e.
min�
max���, �� min�
max � ��������
, ��������
� �. �. ��� 1 (Eq 3)
Each of the above objective functions contains a sub function for each glr3.3 allele. A
minmax optimizer was employed to search the 301-dimensional space for a vector w
that, when the data were projected onto it, minimized the value of the overall objective
function. In the case of Eq. 1, the minimum value of the function would be achieved
when a vector had been found that maximally separated the mutant population means
from the wild type. A t-test was then performed to determine if the mutant means after
projection onto w were significantly different from the wild type but not themselves (Eq.
1); a Brown-Forsythe test was used to determine if the mutant variances were
significantly larger than the wild-type variance (Eq. 2), or if the wild-type variance was
larger than the mutant variances (Eq. 3). The solution vector w for Eq. 1 and an
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equivalent result with explanation obtained with an LDA method similar to Fisher (1936)
without use of a minmax optimizer are shown in Figure S1.
Figure 2 shows the mean values of the results obtained with glr3.3 mutants and
wild type after projection onto the LDA solution vector that minimized Eq. 1 (maximal
separation of mutant and wild-type population means). Conditions B, C, and D
produced statistically significant differences, demonstrating that the gravitropic
responses of the two glr3.3 alleles in these three conditions were not the same as the
wild type. These differences may be considered a growth and development phenotype
for the glr3.3 mutant, albeit one that could not be detected by monitoring the response
of the millimeter-sized root apex without the aid of imaging equipment and computation.
A non-mathematical way to interpret these results is that the distributions of mutant and
wild-type tip angle measurements were not identical. The response of mutant roots to
gravity differed on average from that the wild type as measured morphometrically from
high-resolution image time series.
Figure 3 shows the results of searching for differences in variance (optimizing
Eq. 2) between the mutant and wild-type populations. Normal distributions that best fit
the data are shown along with the actual data points. In all four conditions, glr3.3
populations displayed significantly greater variance than the wild type. In other words,
either glr3.3 allele caused the gravitropic response to be less consistent than the wild
type. Evidence for this was highly statistically significant, while tests for the opposite
effect (Eq. 3, greater variance in the wild type) produced no significant results for three
of the four conditions (data not shown). Interestingly, the optimal solution vectors for
Eq. 1 and Eq. 2 were similarly sinusoidal in shape, though not functionally
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interchangeable (Figure S2). This may indicate that the portions of the response that
the LDA used to separate the means were also the portions in which the mutants varied
more than the wild type. The sinusoidal shape of the solution to Eq. 1 suggested the
next step of analysis.
Figure 4A shows the solution vector (black line) obtained by optimizing Eq. 1
using data from condition C. The shape is reminiscent of a Gaussian distribution
derivative. This raised the possibility that LDA solution vectors satisfying Eq. 1
achieved their separating effects through a property related to the derivative of a
Gaussian distribution. This possibility was further explored using data obtained in
condition C. The two best fitting first-order Gaussian derivative wavelets were found by
a custom algorithm, and are co-plotted (blue dashed lines) with the LDA solution vector
obtained for condition C (Figure 4A). The sum of the two wavelets (solid blue line)
represents a wavelet fit to the solution vector. If this wavelet could separate the mutant
and wild-type population means, the Gaussian derivative components of the solution
vector were probably responsible for its effectiveness in finding a phenotype (Figure
2C). As shown In Figure 4B, the fitted Gaussian derivative wavelet separated mutant
population means from the wild type in condition C similarly to the raw LDA separation
vector, though with lesser statistical significance. This may be expected because the
Gaussian wavelets did not capture all of the features present in the LDA separation
vector. The features not captured probably contributed additionally to the separation of
mutant and wild-type tip angle responses. The fact that the projection values obtained
from the wavelet fit were approximately 3-fold higher compared to the raw LDA
separation vector values is probably due to the fact that the wavelet fit tends to lie
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above the raw separation vector after the 2 h point, increasing the value of each
individual projected onto it relative to the raw vector. The consistency between the
results indicates that the LDA separation vector distinguished the mutant alleles from
the wild type in condition C by acting to a considerable degree as a combination of two
first order Gaussian derivatives.
Convolving a curve with the first derivative of the Gaussian distribution is
common method of obtaining the first derivative of the curve. Thus, the result in Figure
4B could be taken as evidence that the phenotypic difference uncovered by optimizing
solutions to Eq. 1 is actually a difference in the rate of tip angle change at particular
times in the response. This was more directly investigated by performing first and
second-order Gaussian derivative wavelet analysis on the raw data for conditions where
Eq. 1 optimization produced significant solutions (conditions B, C, and D). Wavelets at
scales from 1 to 20 were applied at each point in time to each individual tip angle
response. T-tests of the wavelet-transformed data were performed between each
mutant allele and the wild type to determine how well population means were separated
for each wavelet function tested. After this analysis, significant separation of mutant
population means from the wild type was achieved for condition C but not the others.
Figure 5A shows the original graph of tip angle in response to gravistimulation.
Superimposed on this time course are step functions showing where first order (red) or
second order (blue) Gaussian derivative wavelets significantly separated both mutant
allele populations from the wild type (p < 0.05). In other words, glr3.3 mutations (both
alleles) affected the first derivative, or swing rate (Durham Brooks et al. 2010) when the
red line steps up. The second derivative, or acceleration of the tip angle, differed in
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glr3.3 mutants when the blue line steps down. Putting the two effects together shows
that tip reorientation in glr3.3 plants decelerated significantly more than the wild type
approximately 4.3 h after the onset of gravitropism, when the tip angle was passing
through approximately 40 degrees. From 4.5 h until approximately 7 h, glr3.3 plants
bent more slowly (lower swing rate, first derivative) than wild type. This period was
followed by a brief period during which the wild type decelerated, or 'braked' relative to
the mutants. At the 10-h point, tip angles were closely matched. What the preceding
analysis showed (Figure 5A) is that in condition C the time course by which the root tips
reached the same new steady-state orientation was GLR3.3-dependent.
The phenotypic differences in Figure 5A are statistically robust and consistent
between two independent mutant alleles. Nonetheless, a further test was performed
because variation due to maternal environment can be large and pervasive enough to
affect growth and development of the next generation, especially when measured with
high resolution in seedlings presumably highly dependent on their seed environment.
Therefore, mutant and wild-type seed stocks generated independently of those used in
Figure 5A were assayed in condition C by the same methods. Although the shapes of
the responses in Figure 5B differed from those in Figure 5A (further evidence that
relatively minor maternal effects can significantly affect seedling behavior) the
phenotype was rediscovered by the wavelet analysis. Both glr3.3 alleles braked and
entered a phase of slower swing rate as the tip angle passed through 40 degrees as in
Figure 5A. Again, following this slower response phase, a second-derivative difference
compensated to bring the mutant and wild-type tip angles into close agreement.
Despite substantial differences between the gravitropism time courses displayed by
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seedlings from Generation 1 (Figure 5A) and Generation 2 (Figure 5B) roots, the
acceleration and rate phenotypes were similar in relation to when they developed in the
tip angle time course. Independent generations of two glr3.3 mutant alleles displayed
slower tip swing than wild type as the tip angle passed between 40 and 50 degrees.
DISCUSSION
Gravitropism is a developmental process integral to plant life at least since the
colonization of land. Its facets include environmental signal perception, transduction,
hormone transport, cell expansion, all effected with tight spatial and temporal control
(Blancaflor and Masson 2003; Moulia et al. 2009). Therefore, many genes may be
expected to make small contributions, especially to the modulatory or regulatory
functions. GLR3.3 may be such a gene, for the following reasons. The large, transient
membrane depolarization triggered in wild-type root cells by micromolar levels of amino
acid ligands (Dennison and Spalding 2000) is essentially eliminated by glr3.3 mutations
(Qi et al. 2006; Stephens et al. 2008), as is the large, transient spike in cytoplasmic
Ca2+ that accompanies the depolarization (Qi et al. 2006). So, at the cell-physiological
level, the loss-of-function effects are severe. This is the basis for the proposal that
GLR3.3 is a foundational subunit in multimeric GLR channels present in root and
hypocotyl cells (Stephens et al. 2008). Because genetic redundancy between other
members of the GLR family is not evident in the ionic and electrophysiological assays of
immediate GLR function, redundancy is not a strong explanation for the subtle nature of
the organ-level phenotype described here. However, it is possible that the GLRs affect
growth and development through functions not related to their ion conduction and that
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redundancy in these unknown functions reduce the phenotypic effect of the glr3.3
mutations.
An alternative interpretation of the phenotypes discovered by this application of
machine vision and computation is that channels containing GLR3.3 subunits affect the
stability of the gravitropic response. Without GLR3.3, the response is more variable or
less restrained to develop in a canalized way (Figure 3). Perhaps other growth
responses are similarly less well regulated in glr3.3 mutants so that the proper view of
this gene’s function is as a stabilizer of growth and development. This interpretation
borrows heavily on what has been reported for Hsp90 (Queitsch et al. 2002; Sangster et
al. 2008), and the idea that fundamental mechanisms define the degree of plasticity a
response is permitted (Schlichting and Smith 2002; Schlichting 2008). The role of
plasticity determinants as points of selection and agents of evolutionary change is an
active area of research at the interface of evolution and development (Sultan 2004;
Pigliucci 2005). Another area of research at the other end of the spectrum, intrinsic
noise in gene expression resulting from the low copy numbers of the relevant molecules
per cell (Elowitz et al. 2002) offers a related perspective on how a mutation may cause
little mean phenotype but greater variance in a response. A gene could function to
reduce the intrinsic stochastic component of gene expression in a cell. Mutation of such
a function would be expected to make a cellular response such as coordinated gene
expression more variable, not much affect the mean, but nonetheless have natural
selection consequences (Çağatay et al. 2009).
If growth and development were more routinely measured with high resolution
and in multiple conditions, the frequent conclusion that a mutant has no phenotype may
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be replaced by the finding of a defect in modulation or regulation of a process. Such
phenotypes may appear minor or unimportant when observed in the laboratory and
considered singly. However, growth and development may be better thought of as a
process that depends on hundreds or thousands of such modulatory effects that
integrate to confer the appropriate degree of response plasticity in evolutionarily
relevant scenarios.
Regardless of whether the effects described here represent the largest or the
smallest contributions to growth and development to be discovered for GLR3.3, they
add some insight into the root gravitropism mechanism. The wavelet analysis
demonstrated that GLR3.3 promoted curvature development after the tip angle reached
40 degrees. Previous research demonstrated that maximum swing rate occurs at a tip
angle of approximately 30 degrees, regardless of condition or overall response time
course (Durham Brooks et al. 2010). Following this maximum, tip angle rapidly
decelerates as part of autotropic straightening, which counteracts gravitropic signaling
so that the reoriented portion of the root begins to grow straight (Stankovic et al. 1998).
The glr3.3 phenotype is detected soon after this event, indicating that this gene may act
to counter or buffer against the straightening response.
Some ionic and electrophysiological events have been observed to follow gravity
stimulation (Lee et al. 1983; Scott and Allen 1999; Plieth and Trewavas 2002; Massa et
al. 2003). Of them, only the rapid change in cytoplasmic pH in the gravity sensing cells
of the apex has been causally linked to the ensuing growth/curvature response (Fasano
et al. 2001; Hou et al. 2004). Possibly, GLR3.3 and other family members generate
ionic events in response to gravity that relate more to response modulation as
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established here than creation of the differential growth responsible for tip bending. The
present results may prove helpful in directing cell physiology studies to the time and
place when gravity-induced ionic phenomena related to response modulation and
dependent on GLR3.3 may be found.
The method described here will be most valuable when used to generate
quantitative descriptions of large numbers of mutants that can be mapped onto each
other over the course of a developmental process such as gravitropism. Machine
learning methods could be used to classify the LDA results of different mutants to find
functional relationships between genes even if visible phenotypes are not present or
draw the attention in a different direction. A similar approach has been used in C.
elegans to classify locomotive behavior of a subset of mutants involved in nervous
system function (Geng et al. 2003). If widely adopted in Arabidopsis research, the
approach used here would result in a much larger fraction of today’s mutant populations
being useful to the process of discovering gene function.
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FIGURE LEGENDS
Figure 1. Change in root tip angle of wild type and two glr3.3 mutants after
gravistimulation in four conditions: (A) Two-day-old seedlings from small seeds, 15 ≤ n
≤ 23 (B) Two-day-old seedlings germinated from large seeds, 18 ≤ n ≤ 27 (C) Four-day-
old seedlings germinated from small seeds, 15 ≤ n ≤ 23, and (D) Four-day-old seedlings
germinated from large seeds, 14 ≤ n ≤ 22. Tip angle was automatically measured from
high-resolution time series of root images bending toward gravity using custom image
analysis software. Error bars are standard error of the mean.
Figure 2. Time course data of root tip angle for each individual was plotted in a higher-
dimensional space, allowing for each individual to be represented as one point in 301-
dimensions. Linear discriminant analysis was used to find a vector such that when the
individuals were projected onto it, it maximally separated the mutant means from the
wild type. Population means are shown after projection of individual gravitropic
responses onto the linear solution of Eq 1 optimization. (A) Two-day-old seedlings
germinated from small seeds. (B) Two-day-old seedlings germinated from large seeds,
(C) Four-day-old seedlings germinated from small seeds, and (D) Four-day-old
seedlings germinated from large seeds. Asterisks indicate significant differences in
population means as determined by a two-sample t test. **, p < 0.01; ***, p < 0.001.
Figure 3. Population variances after projection of individual gravitropic responses onto
the linear solution of Eq. 2 optimization. Shown is the normal distribution that best fits
the projection data and below each normal fit are the points corresponding to each
individual within the population. (A) Two-day-old seedlings germinated from small
seeds, (B) Two-day-old seedlings germinated from large seeds, (C) Four-day-old
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seedlings germinated from small seeds, and (D) Four-day-old seedlings germinated
from large seeds. Pound symbols indicate significant differences in population variance
between mutant and wild-type as determined by a Brown-Forsythe test. ##, p < 0.01;
###, p < 0.001.
Figure 4. Describing a mean-separating LDA solution vector as a composition of first-
order Gaussian derivative wavelets. (A) An LDA solution vector for Eq 1 optimization in
condition C is shown in black. The two first-order Gaussian wavelets that best fit the
solution vector are shown with dashed blue lines. Summing these two wavelets gives
the solid blue line, which represents a wavelet fit of the LDA solution vector. (B) The
individual responses from condition C were projected onto the LDA solution vector (left
three bars) or onto the wavelet fit (solid blue line from panel A). A two-sample t test
determined that the wavelet fit significantly separated the population means. * p <0.05;
** p < 0.01; *** p < 0.001
Figure 5: Derivative analysis of two independent generations of glr3.3 alleles
responding to gravistimulation in condition C (four-day-old roots from small seeds).
Shown are the average tip angle responses to gravistimulation for wild type (black) and
the mutant alleles (orange and light orange). Error bars show the standard error of the
mean. (A) Generation 1 of mutant and wild-type seed stocks. (B) Generation 2 of
mutant and wild-type seed stocks. n ≥ 8 for all populations. Regions of the time
courses in which first-order Gaussian derivative wavelets significantly separated the
mutant populations from the wild type are indicated by an upward deflections in the red
line (p < 0.05 as determined by a two-sample t test). Downward deflections of the blue
line indicate portions of the time courses in which second-order Gaussian wavelets
22
significantly separated the mutant populations from the wild type (p < 0.05 as
determined by a two-sample t test).
23
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ACKNOWLEDGMENTS
This work was supported by NSF grant DBI-0621702 and DOE grant DE-FG02-
04ER15527 to EPS, and an NSF graduate fellowship to TLDB.