Lily Pollen Tube Cell Wall Growth
Sylvester McKenna,
Larry Winship,
Research in the lab of Peter Hepler
Math by Joe Kunkel
Pollen tube auto-peak-identification.
1. Growing tubes have 200+ image frames with
600+ pixel length. Automation is a priority.
2. The tube tip profile does not always own the
maximum density on the profile.
3. The profiles do not always have two distinct
peaks associated with the outer and inner edge.
How does one deal with shoulder peaks.
4. The outer base density differs from the inner
base density.
Pollen tube axial profile tip ID:
True tip profile.
• In some tubes the maximum fringe is not the tip profile!
• Auto-identification of tip profiles must deal with all cases.
Spurious peak.
True tip profile.
R scripting:
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Wall_1_Prep.R
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Wall_2_dGauss.R
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Initial peak identification allows 2 peaks to
be given a peak pixel location.
Peak refinement allows fractional pixel
identification by fitting two Gausian curves
to the DIC profile of the pollen tube tip wall.
Shoulders on primary peaks can be
identified as a Gaussian component.
Wall_3_Anal.R
–
Wall-thickness and Tip Velocity crosscorrelation analysis.
R script 1: Wall_1_Prep.R menu …
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back | <
fwd | >
98% dx | 102% dx | +
tog-5 | 5
+2nd | 2
-2nd | 1
no_force | 0
goto | g
browse | b
cntr | c
sort | s
wid | w
nar | n
abort | *
sets previous 10 profiles as current.
set next 10 profiles as current.
decrease the Y-axis focus span.
increase the Y-axis focus span.
shift span of profiles by 5.
add or re-add a second peak.
remove a second peak.
remove force on choosing a peak.
go to any frame in the series.
allows examining current variables.
allows choice of a new focus for a frame.
does a sort of the peak orders for current.
widen the X-axis pixel span.
narrow the X-axis pixel span.
close down the program.
R script 1: Wall_1_Prep.R I/O
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Input
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Requests the name of a ‘.log’ file.
Asks if the log file is new.
If not new, asks for a numeric suffix of prior output.
Asks if an output in memory should be used/recovered.
Asks for a numeric suffix to add to an output file.
• A menu allows character driven choice of changes to identifying
pollen tube tip profile parameters:
– Identify a proximal profile peak pixel.
– Review auto-chosen peaks and allow changes.
• Produces an output file identified by a numeric suffix:
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Cols 2-3: peak-1 and peak-2 X-axis location in um.
Cols 9,11: peak-1 and peak-2 integer pixel location.
Cols 10,12: peak-1 and peak-2 standard deviation (1 by default)
Col 7: 0, 1, 2 auto peaks or 3, a forced set of peaks.
Col 8: 0, 1 indicates whether current means and SD are set.
Col 4,5,6: F, D , Diff parameters for mixing Gaussian distributions.
R-script 2: Wall_2_dGauss.R
• Helps fit a pair of
Gaussian distributions
to the pollen tube DIC
profile.
• Menu allows changes
to mean and SD of
leading and trailing
Gaussian distributions.
• Ends with using
means to estimate tip
velocity and wall
thickness.
R script 2: Wall_2_dGauss.R I/O
•
Input
–
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–
–
Requests the name of a ‘.log’ file.
Asks for a numeric suffix of prior output.
Asks if an output in memory should be used/recovered.
Asks for a numeric suffix to add to an output file.
• A menu allows character driven choice of changes to set pollen tube
tip profile parameters:
– Identify profile peaks that need mean, SD, F and D adjustment.
– Allows inheritance of SD, F and D parameters from previous profile.
– Sets Col 8 to 1 when adjustments have been made.
Contour
prior to
fitting
• Produces an output file identified by a numeric suffix:
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Cols 2-3: peak-1 and peak-2 X-axis location in um.
Cols 9,11: peak-1 and peak-2 integer pixel location.
Cols 10,12: peak-1 and peak-2 standard deviation (1 by default)
Col 7: 0, 1, 2 auto peaks or 3, a forced set of peaks.
Col 8: 0, 1 indicates whether current means and SD are set.
Col 4,5,6: F, D , Diff parameters for mixing Gaussian distributions.
• Each contour must be adjusted individually, thus a need for
efficiency.
Contour
after
fitting
Protocol for fitting contours to a tip profile:
• The means, u1 and u2, are preliminarily set
as the center of the pixel identified as the
peak of a DIC fringe. The SDs are
preliminarily set to 1 for convenience.
• Means, u1 and u2, and SDs are changeable
in adjustable sub-pixel increments.
• The joint contour is auto adjusted to
coincide with the peak of the contour data
with parameter F for visual display.
• The relative contribution of peak 1 and 2 to
the joint contour is adjustable in increments
by parameter D to visually conform to data.
• u1, SD1, u2, SD2, F and D are stored in an
output matrix which can reproduce the
predicted outline.
• SDs, F and D are inheritable as an aid.
• Wall Thickness is computed as u2-u1. Tip
Velocity is computed from the positions of
u1 in adjacent frames.
Contour
prior to
fitting
Contour
after
fitting
Sample output of Wall_dGauss.R:
Wall_3_Anal.R: Wall thickness and Tip Velocity
Sample output of Wall_Anal.R:
• Original data fit with lowess(f=0.02)
• Corrected by subtracting median with lowess(f=0.2)
Wall_3_Anal.R: Wall thickness and Tip Velocity
…for data set 061506f …
Sample output of Wall_Anal.R:
A.
B.
• A. Auto-correlation of tip-velocity.
• B. Cross-correlation of Tip-Velocity with Wall Thickness.
• Shows that Tip Velocity follows Wall-Thickness peak but predicts the
Wall-Thickness trough better. Is that significant and meaningful?
• Are these conclusions generalizeable to other data sets?
Wall_3_Anal.R: Wall thickness and Tip Velocity
…offset correlations for data set 061506f …
A.
Sample output of Wall_Anal.R:
B.
• A. Correlation of Tip-velocity preceding Wall thickness by 3-7 frame units.
• B. Correlation of Tip-Velocity after Wall Thickness by 3-7 frame units.
• Shows that Tip Velocity predicts Wall-Thickness, r2 = 0.44, but predicts the WallThinness a bit better, r2= 0.59. Is that significant and meaningful?
• Are these conclusions generalizeable to other data sets?
Andy.R: Wall thickness and Tip Velocity
…ANOVA of set 061506f …
Sample output of Andy.R:
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Individually Velocity is correlated with Wall Thickness (-7 and +6) by (r = 0.66, -0.77).
Using both provides a better joint correlation of r = 0.80.
Wall-Thickness predicts Tip Velocity, r2= 0.64 which is better than the individual
explanations (r2 = 0.44, 0.59). This is significant and meaningful!
Are these conclusions generalizeable to other data sets?
Andy.R: Wall thickness and Tip Velocity
…ANOVA of set 061506f …
Output of Andy.R:
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Individually Velocity is correlated with Wall Thickness (-7 and +6) by (r = 0.66, -0.77) or
(r2 = 0.44, 0.59).
Using both provides a better joint correlation of r = 0.80 or 64% of variability.
The effects are not visually or dramatically but are statistically significant.
Are these conclusions generalizeable to other data sets? (now have two sets consistent.)
How Andy.R and was initially used
• Wall_3_Anal.R provided an output
matrix: 061506fs.log2.out.csv of
Velocity plus offset Wall Thickness
data. The original Nframes of data
needed to be trimmed by 7 lines at
beginning and end to allow for
offsets of the wall data in both
directions.
• This data with a column of 1’s
representing the mean was submitted
to Andy.R and the equation V = u +
Wth(-7) + Wth(+6) was evaluated.
• The contribution of the two WallThickness offsets to predicting V
were tested by subtraction and found
to be both highly significant.
• The parameters of the
equation were used to
predict V and the result
was found to explain
64% of the variability of
V.
• Eventually 6-tubes data
would be analyzed
together.
GAM analysis 1
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Formula: Velocity ~ s(Wp3) + s(Wp4) + s(Wp5) + s(Wp6) + s(Wp7) + s(Wm3) + s(Wm4) + s(Wm5) + s(Wm6) +
s(Wm7)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001956 0.002167 0.903 0.368
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Approximate significance of smooth terms:
edf Est.
rank
F
s(Wp3)
1.000
1
6.027
s(Wp4)
2.327
5
3.053
s(Wp5)
1.064
3
1.597
s(Wp6)
1.000
1
4.210
s(Wp7)
1.672
4
1.926
s(Wm3)
2.038
5
1.422
s(Wm4)
3.597
8
2.752
s(Wm5)
1.000
1
1.257
s(Wm6)
1.000
1
0.023
s(Wm7)
5.487
9
2.133
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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R-sq.(adj) = 0.75 Deviance explained = 77.4%
GCV score = 0.0011155 Scale est. = 0.001005 n = 214
Conclusion: Focus on the significant offsets (-7, -4, 3, 4, 6)!
p-value
0.01498 * <- {1 df thus it does not need smooth)
0.01123 * <0.19140
0.04155 * <- {1 df thus it does not need smooth)
0.10772
0.21795
0.00672 **<0.26368
0.88051
0.02855 * <-
GAM analysis 2
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Formula: Velocity ~ s(Wp3) + s(Wp4) + s(Wp6) + s(Wm4) + s(Wm7)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001956 0.002197 0.891 0.374
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Approximate significance of smooth terms:
edf
Est.rank F
p-value
s(Wp3)
1.000
1
6.943
0.00908 ** {1 df thus it does not need smooth)
s(Wp4)
2.531
6
3.245
0.00458 **
s(Wp6)
1.052
3
8.940
1.39e-05 *** { ~1 df thus it does not need smooth?)
s(Wm4)
4.010
9
5.753
4.23e-07 ***
s(Wm7)
4.921
9
2.508
0.00968 **
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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R-sq.(adj) = 0.743 Deviance explained = 75.9%
GCV score = 0.0011079 Scale est. = 0.0010327 n = 214
GAM analysis 3
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Formula: Velocity ~ s(Wp6) + s(Wm7)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001956 0.002581 0.758 0.449
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Approximate significance of smooth terms:
edf
Est.rank F
p-value
s(Wp6)
2.080
5
25.57
< 2e-16 ***
s(Wm7)
1.001
2
16.68
1.89e-07 ***
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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R-sq.(adj) = 0.645 Deviance explained = 65%
GCV score = 0.0014538 Scale est. = 0.001426 n = 214
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The model : Velocity ~ s(Wp6) + s(Wm7) + s(Wp6,Wm7) yielded no evidence of
an interaction between the leading peak of Wall-Thickness and trailing trough.
GLM and GAM correlations
Data set 061506f:
•R
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Vo
Ve
pG
pG2
pG3
• R2
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Vo
Ve
pG
pG2
pG3
Vo
Ve
pG
pG2
pG3
1.000
0.844
0.880
0.871
0.806
0.844
1.000
0.975
0.972
0.944
0.880
0.975
1.000
0.993
0.930
0.871
0.972
0.993
1.000
0.934
0.806
0.944
0.930
0.934
1.000
Vo
Ve
pG
pG2
pG3
1.000
0.713
0.774
0.759
0.650
0.713
1.000
0.951
0.945
0.891
0.774
0.951
1.000
0.987
0.865
0.759
0.945
0.987
1.000
0.873
0.650
0.891
0.865
0.873
1.000
all offsets
(-7, -4 , 3, 4, 6)
Wall thickness and Tip Velocity
… / / GLM on set 061506f … then GAM /
Output of GAM /\ vs GLM /\ /\ :
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Individually Velocity is correlated with Wall Thickness, (-7 and +6) by (r = 0.66, -0.77).
Using both provides a better joint correlation of r = 0.80 (r2=0.64) above in red.
Knowledge of Wall Thickness explains substantial amounts of the variation of Tip Velocity.
The general additive model (GAM) achieves an r =0.88 (r2=0.77) using 10 Wall
Thickness offsets. An r =0.87 (r2=0.76) using 5 Wall Thickness offsets (-7, -4 , 3, 4, 6).
Replicates of Corr 1, 2:
Replicates of Corr 3, 4:
Replicates of Corr 5, 6:
Analysis of 6 tube data via GLM
1.
Velocity and Wall-thickness offset data from
6 tubes were combined into one data +
design matrix.
Analysis of dispersion script andy.R was
used to fit Velocity to Wall-Offset and
pollen-tube specific effects:
Vmi = u + WTpj + WTmk + tubem
2.
3.
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4.
5.
i = image frame 1…Nm
j = Wall thicknesses offset plus direction
k = Wall thicknesses offset mins direction
m = Pollen tube #, 1…6.
Nm = Number of image frames in tube m.
Since these items are all correlated, they
must be analyzed by subtraction from a joint
reduction of Velocity Sums of Squares.
Wall thickness as measured physically and
by PI fluorescence can each be measured in
what sense they can predict Tip Velocity.
Tube data sample sizes:
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>
t(Xdes[N1,12:16])%*%Xdes[N1,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
210
0
0
0
0
Tube2
0
0
0
0
0
Tube3
0
0
0
0
0
Tube4
0
0
0
0
0
Tube5
0
0
0
0
0
>
t(Xdes[N2,12:16])%*%Xdes[N2,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
0
0
0
0
0
Tube2
0
187
0
0
0
Tube3
0
0
0
0
0
Tube4
0
0
0
0
0
Tube5
0
0
0
0
0
>
t(Xdes[N3,12:16])%*%Xdes[N3,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
0
0
0
0
0
Tube2
0
0
0
0
0
Tube3
0
0
210
0
0
Tube4
0
0
0
0
0
Tube5
0
0
0
0
0
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
>
t(Xdes[N4,12:16])%*%Xdes[N4,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
0
0
0
0
0
Tube2
0
0
0
0
0
Tube3
0
0
0
0
0
Tube4
0
0
0
210
0
Tube5
0
0
0
0
0
>
t(Xdes[N5,12:16])%*%Xdes[N5,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
0
0
0
0
0
Tube2
0
0
0
0
0
Tube3
0
0
0
0
0
Tube4
0
0
0
0
0
Tube5
0
0
0
0
210
>
t(Xdes[N6,12:16])%*%Xdes[N6,12:16]
Tube1 Tube2 Tube3 Tube4 Tube5
Tube1
210
210
210
210
210
Tube2
210
210
210
210
210
Tube3
210
210
210
210
210
Tube4
210
210
210
210
210
Tube5
210
210
210
210
210
N1= 210, N2= 187, N3= 210, N4= 210, N5= 210, N6= 210
Design Matrix X: Independent Vars
ß
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[1,]
[2,]
[3,]
[4,]
[5,]
[6,]
[7,]
[8,]
[9,]
[10,]
[11,]
[12,]
[13,]
[14,]
[15,]
[16,]
[17,]
[18,]
[19,]
[20,]
Mean WTp3 WTp4 WTp5 WTp6 WTp7 WTm3 WTm4 WTm5 WTm6 WTm7 Tube1 Tube2 Tube3 Tube4 Tube5
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-68
-41
-10
18
50
29
78
113
172
130
71
102
84
102
146
99
58
42
15
-10
-41
-10
18
50
29
78
113
172
130
71
102
84
102
146
99
58
42
15
-10
-12
-10
18
50
29
78
113
172
130
71
102
84
102
146
99
58
42
15
-10
-12
-31
18
50
29
78
113
172
130
71
102
84
102
146
99
58
42
15
-10
-12
-31
-62
50
29
78
113
172
130
71
102
84
102
146
99
58
42
15
-10
-12
-31
-62
-5
-44
-14
-53
-67
-76
-74
-68
-41
-10
18
50
29
78
113
172
130
71
102
84
102
Velocity = X ß + error
-34
-44
-14
-53
-67
-76
-74
-68
-41
-10
18
50
29
78
113
172
130
71
102
84
31
-34
-44
-14
-53
-67
-76
-74
-68
-41
-10
18
50
29
78
113
172
130
71
102
-11
31
-34
-44
-14
-53
-67
-76
-74
-68
-41
-10
18
50
29
78
113
172
130
71
43
-11
31
-34
-44
-14
-53
-67
-76
-74
-68
-41
-10
18
50
29
78
113
172
130
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
E[Velocity] = X ß
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
R-prompt plotting of 6 Velocity curves
> par(mar=c(0.1, 0.1, 0.1, 0.1) + 0.1) # reduces margins
> layout(matrix(c(1:6), 6, 1, ))
# creates multi-plot layout
> plot(Velocity[N1], typ='p')
# plot tube 1 data
> lines(EVel[N1], typ='l', lwd=3) # add expected tube 1 line
> plot(Velocity[N2], typ='p')
#…
> lines(EVel[N2], typ='l', lwd=3)
> plot(Velocity[N3], typ='p')
> lines(EVel[N3], typ='l', lwd=3)
> plot(Velocity[N4], typ='p')
> lines(EVel[N4], typ='l', lwd=3)
> plot(Velocity[N5], typ='p')
> lines(EVel[N5], typ='l', lwd=3)
> plot(Velocity[N6], typ='p')
> lines(EVel[N6], typ='l', lwd=3)
E[Velocity] = X ß, R = 0.749, R2 = 0.561
Tip velocity is fit by a common function of prior and succeeding wall-thicknesses.
061406q
061506c
061506e
061506f
061506l
061506m
Analysis of 6 tube data via GAM
1.
Velocity and Wall-thickness offset data from
6 tubes were combined into one data +
design matrix.
require(mgcv):
Vmi ~ s(WTpj) + s(WTmk) + tubem
2.
3.
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•
•
4.
5.
6.
7.
i = image frame 1…Nm
j = Wall thicknesses offset in plus direction
k = Wall thicknesses offset in minus direction
m = Pollen tube #, 1…6.
Nm = Number of image frames in tube m.
Since these items are all correlated, they
must be analyzed by subtraction from a joint
reduction of Velocity Sums of Squares.
Wall thickness as measured physically and
by PI fluorescence can each be measured in
what sense they can predict Tip Velocity.
There is no significance to different pollen
tubes in predicting tip velocity in either
GLM or GAM analysis!! Ergo, it is a very
uniform process!!!
Rscript to compare GAM to GLM fit:
source("Velo-WTglm+WTgam.R")
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Formula: Velocity ~ s(WTp3) + s(WTp4) + s(WTp5) +
s(WTp6) + s(WTp7) + s(WTm3) + s(WTm4) + s(WTm5) +
s(WTm6) + s(WTm7) + Tube1 + Tube2 + Tube3 + Tube4 +
Tube5
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001402 0.001489 0.942 0.346
Tube1
-0.003875 0.003468 -1.118 0.264
Tube2
0.003306 0.003983 0.830 0.407
Tube3
0.002695 0.003435 0.785 0.433
Tube4
-0.003875 0.003468 -1.118 0.264
Tube5
0.001558 0.003357 0.464 0.643
Approximate significance of smooth terms:
edf Est.rank F p-value
s(WTp3) 5.040
9 5.775 6.93e-08 ***
s(WTp4) 5.164
9 4.150 2.77e-05 ***
s(WTp5) 1.000
1 6.281 0.01234 *
s(WTp6) 4.348
9 2.049 0.03131 *
s(WTp7) 7.188
9 2.533 0.00704 **
s(WTm3) 6.520
9 4.826 2.37e-06 ***
s(WTm4) 1.000
1 1.780 0.18238
s(WTm5) 5.287
9 2.620 0.00533 **
s(WTm6) 8.557
9 4.061 3.80e-05 ***
s(WTm7) 5.922
9 3.009 0.00148 **
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
•
•
R-sq.(adj) = 0.601 Deviance explained = 61.8% R = 0.786
GCV score = 0.002865 Scale est. = 0.0027353 n = 1237
E[Velocity] = GAM(X), R = 0.786, R2 = 0. 618
Tip velocity is fit by GAM(wall-thicknesses+tube#) and compared to GLM.
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Notes on R-scripts
Fixes of Wall R-scripts 0.9:
1.
2.
3.
4.
Added memory of a recent log file.
Added output of a CSV file of Velocity and
offset Wall thicknesses used to analyze the
prediction of Velocity using andy.R and
GAM.
Problem with sort routine when reloading
an output file. Needed to be loaded
as.matrix() rather than as a data frame.