Tittlke
Supplemental Data S1
Title:Mathematical modeling-guided evaluation of the
biochemical, developmental, environmental and genotypic
determinants of essential oil composition and yield in peppermint
leaves
Authors:Rigoberto Rios-Estepa, Iris Lange, James M. Lee, and B.
Markus Lange
INDEX
Page
1. Estimation of enzyme concentrations in individual glandular
trichomes
1
2. Estimating developmental dynamics of glandular trichome
density
6
3. Kinetic properties of enzymes involved in peppermint
monoterpene biosynthesis
7
4. Generating a system of ordinary differential equations to
describe kinetic properties of enzymes 9
5. Calculating monoterpenoid essential oil yields for individual
glandular trichomes
11
6. Estimating changes in enzyme concentrations based on
measurements of gene expression levels12
7. Kinetic model for peppermint monoterpene biosynthesis
14
8. Statistical analysis of goodness of fit between simulated and
measured monoterpene profiles26
1.Estimation of enzyme concentrations in individual glandular
trichomes
Peppermint glandular trichomes harbor three different cells
types termed basal, stalk and secretory cells (Scheme 1). Among
these cell types the secretory cells (eight-celled disk highlighted
in Scheme 1; only four cells are visible in cross section) are
responsible for the biosynthesis of monoterpenes in peppermint. To
allow the calculation of enzyme concentrations in the secretory
cells (a prerequisite for building Michaelis-Menten-based kinetic
models), the volume of these specialized cells had to be
determined.
Scheme 1
Enzyme activity
[
µ
mol/h/leaf]
a
1
a
2
a
3
c
1
c
2
c
3
b
1
b
2
b
3
0
10
0
10
20
30
40
50
60
Leaf age [d]
a
1
a
2
c
1
c
2
b
1
b
2
b
3
Enzyme activity
[
µ
mol/h/leaf]
a
1
a
2
a
3
c
1
c
2
c
3
b
1
b
2
b
3
0
10
0
10
20
30
40
50
60
Leaf age [d]
a
1
a
2
c
1
c
2
b
1
b
2
b
3
1.1Determining the volume of secretory cell clusters
The shape of the secretory cell cluster was approximated by a
frustum of a cone. The height of the frustum and the relevant radii
were determined experimentally by analyzing 20 microscopic images
of peppermint leaf cross sections (average values are given below).
The volume of the secretory cell cluster was thus calculated as
V(secretory cell cluster) = 1/3 π h (R2 + R r + r2) = 2.38 x
10-5 µL
where
h = height of frustum = 15 µm
r = radius of frustum at narrower end = 14 µm
R = radius of frustum at wider end = 30 µm (diameter is 60 µm as
indicated in Scheme 1)
The volume of an individual secretory cell was obtained as 1/8
of the volume of the entire secretory cell cluster:
V(secretory cell) = 2.74 x 10-5 µL x 1/8 = 2.98 x 10-6 µL
1.2Estimating the volume desities of subcellular
compartments
Enzymes involved in peppermint monoterpene biosynthesis are
localized to specific subcellular compartment. This means that for
the purpose of developing a model, the actual concentration in the
appropriate compartment needs to be determined. The volumes of
organelles within a secretory cell were estimated by two different
approaches using ImageJ software (http://rsb.info.nih.gov/ij). The
percentage of micrograph area (volume density) covered by
leucoplasts and mitochondria was directly calculated by encircling
organelles with a calibrated measuring tool. Such direct
measurements using the ImageJ drawing tool were feasible for larger
organelles, but were impractical for the tubular smooth ER, which
consists of numerous interconnected tubes with narrow diameters.
The volume densities of plastids, mitochondria, ER, vacuoles and
cytosol were also determined by randomly superimposing a
stereological grid overlay on gland cell micrographs and by
counting the number of intercepts these organelles made with test
points. The grid overlay plug-in for ImageJ was obtained at
http://rsb.info.nih.gov/ij/plugins/grid.html. The test points
consisted of the intersections of horizontal and vertical lines
(the corners of grid squares) separated by spacing representing 1
µm on the micrographs.
Representative micrographs of secretory cells from six different
secretory-phase peltate glandular trichomes were used for the
morphometric measurements. These were chosen to include both apical
and basal regions of gland cells, since there is some polarity in
the distribution of organelles [3]. All specimens were preserved by
high-pressure freezing and freeze-substitution in order to ensure
good preservation of gland cell ultrastructure. However, this
method leads to an extraction of low molecular weight lipids during
the freeze-substitution process, so that stereological estimates of
volume density for vacuolar and cytoplasmic monoterpene droplets
could not be obtained.
The directly determined volumes for plastids and mitochondria
were very similar to those obtained with volume densities
calculations from test-point counts, confirming the quality of our
stereological estimates. The average volume density for leucoplasts
was 13.3 % area (directly measured) and 13.9 % (stereological
test-point intercepts). The difference between these values (0.6 %)
is smaller than the standard deviation between leucoplast volume
densities for the six individual glandular trichomes (σVd = 3.25
directly measured; σVd = 4.02 when determined using stereological
methods). The difference between the estimates for the
mitochondrial volumes based on these methods (1.0 %) is slightly
larger than the standard deviations between individual glandular
trichomes (σVd = 0.6 directly measured; σVd = 0.8 when determined
using stereological methods).
Table 1. Volume densities of subcellular compartments in
peppermint glandular trichomes.
Organelle
Fraction of cross-sectional area [%]
σVd
Estimated volume per secretory cell
[µL]
Average diameter
[µm]
σdi
Surface area per secretory cell [µm2]
Leucoplasts
13.9
4.02
0.41x 10-6
n.m.
---
---
Mitochondria
4.4
0.6
0.13 x 10-6
0.47 (n=95)
0.11
1.66 x 103
ER
36.5
3.5
1.07x 10-6
0.07 (n=265)
0.04
1.53x 104
Vacuoles
16.2
4.3
0.48 x 10-6
n.m.
---
---
Cytoplasm
20.4
3.1
0.60x 10-6
n.m.
---
---
Other
8.6
---
---
---
---
---
Definitions: σVd, standard deviation of volume density; σdi,
standard deviation of average diameter.
The number of point counts required to obtain accurate volume
densities were calculated according to published methods (Weibel
(1979) in Stereological Methods, Vol 1, Practical Methods for
Biological Morphometry, ed Weibel ER (Academic Press, London),
63–100). The required number of point counts (Pc) is proportional
to the standard error and is reduced with larger organelle volume
density (Vva) and with a larger number of replicate specimens (m)
by the equation
Pc = (tα2/(m d2)) ((1-Vva)/Vva))
where
d = Confidence interval (standard error of the mean)
m = Number of replicates
tα = Acceptable error probability
VVa = Organelle volume density
For example, the required number of point counts (per glandular
trichome) to obtain a 95 % probability for accuracy within a 10 %
confidence interval would be 409.5 (total of 2457 counts) for
leucoplasts. Our actual count consisted of 2,266 test-point counts
with an average of 378 counts per specimen (with a range of 233 to
476), which results in estimated confidence intervals at the 95%
probability level of 10.4 % for leucoplasts, 17.6 % for
mitochondria, 9.6 % for vacuoles, and 5.5 % for ER. We also
calculated the total surface area of mitochondria and ER within
secretory cells by assuming that mitochondria are spherical and the
smooth ER consists of narrow cylinders.
1.3Estimating maximum enzyme concentrations in secretory phase
glandular trichomes
The laboratory of Rodney Croteau at Washington State University
succeeded in cloning the genes corresponding to all enzymes
directly involved in the p-menthane pathway of monoterpene
biosynthesis in peppermint. Individual genes were expressed in
appropriate heterologous expression vectors. The corresponding
recombinant enzymes were purified to apparent homogeneity and used
to generate highly specific antibodies. Enzyme concentrations were
determined by Western Blotting with enzyme extracts obtained at the
time of the highest biosynthetic activity. The subcellular
localization of monoterpene biosynthetic enzymes was determined by
immunocytochemistry with the same antibodies. An overview of these
studies is provided in a recent review article (Croteau et al.
(2005) Naturwissenschaften 92: 562-577).
Table 2. Maximum concentrations of monoterpene biosynthetic
enzymes in peppermint secretory cells.
_______________________________________________________________________________________
Enzyme
Reference
Concentration in
secretory cells [µM]
_______________________________________________________________________________________
Geranyl diphosphate synthase
Turner and Croteau (2004) Plant Physiol. 136: 4215
0.030000
(-)-Limonene synthase
Turner et al. (1999) Plant Physiol. 120: 879
0.017000
(-)-Limonene 3-hydroxylase
Turner and Croteau (2004) Plant Physiol. 136: 4215
0.003000
(-)-trans-Isopiperitenol dehydrogenaseTurner and Croteau (2004)
Plant Physiol. 136: 4215
0.900000
(-)-Isopiperitenone reductase
Turner and Croteau, unpublished data
0.340000
(+)-cis-Isopulegone isomerase
Turner and Croteau, unpublished data
0.340000
(+)-Menthofuran synthase
Bertea et al. (2001) Arch. Biochem. Biophys. 390: 279
0.000070
(+)-Pulegone reductase
Turner and Croteau (2004) Plant Physiol. 136: 4215
0.001500
(-)-Menthone:(-)-menthol reductaseTurner and Croteau,
unpublished data
0.001100
(-)-Menthone:(+)-neomenthol reductaseTurner and Croteau,
unpublished data
0.000011
_______________________________________________________________________________________
1.4Developmental patterns of monoterpene biosynthetic enzyme
activities
For the majority of kinetic mathematical models it is assumed
that the amounts of biosynthetic enzymes remain constant for the
duration of the experimental period. In peppermint glandular
trichomes, the biosynthesis of monterpenes involves dynamic changes
in the activities of biosynthetic enzymes (Gershenzon et al. (2000)
Plant Physiol. 122: 205-214; McConkey et al. (2000) Plant Physiol.
122: 215-224). Genes and enzymes involved in The maximum amount of
each enzyme present at the peak of monoterpene biosynthesis (15 d
for most enzymes; 20 d for (-)-menthone:menthol reductase) was
determined based on immunochemical data as described in 1.3. We
thus used the available experimental data on developmental changes
in biosynthetic enzyme activities to approximate changes in enzyme
amounts with a Gaussian function:
A) Experimental enzyme activity data
B) Example of a Gaussian function to approximate enzyme activity
data
Table 6. Extrapolation of protein levels from experimentally
determined gene expression levels.
Enzyme WT - GH^WT - LL
Z=100; W=0.05; GN(max)=10151; S0=1900Z=1500; W=0.05;
GN(max)=7004; S0=950
Gene Exp.Strd. [Enz] in theGene Exp.Protein Exp.[Enz] in theGene
Exp.Protein Exp.[Enz] in theGene Exp.Protein Exp.[Enz] in the
LevelErrormodel [
µ
M]*vs. WT-GHvs. WT-GHmodel [
µ
M]vs. WT-GHvs. WT-GHmodel [
µ
M]vs. WT-GHvs. WT-GHmodel [
µ
M]
DXS
#29.911.350.030000.560.820.024621.121.030.030860.660.870.02612
DXR7.040.520.022500.310.640.014471.901.190.026721.541.120.02527
CMK6.192.210.022500.560.820.018472.261.240.027881.261.060.02395
HDS28.621.350.500000.580.830.414722.421.260.630203.181.340.67108
LS5.181.890.017000.730.900.015301.471.110.018881.621.140.01937
L3H7.182.180.003002.641.290.003860.400.720.002153.841.400.00420
PR0.170.820.001509.501.670.0025114.281.790.002691.711.160.00173
MFS16.661.850.000073.301.350.000092.101.220.000096.351.550.00011
Enzyme MFS7a - GHMFS7a - LWMFS7a - LLMFS7a - LL/HT
Z=10; W=0.05; GN(max)=12382; S0=1900Z=200; W=0.05; GN(max)=7593;
S0=1500Z=400; W=0.05; GN(max)=$; S0=1100 Z=400; W=0.05;
GN(max)=5920; S0=950
Gene Exp.Protein Exp.[Enz] in theGene Exp.Protein Exp.[Enz] in
theGene Exp.Protein Exp.[Enz] in theGene Exp.Protein Exp.[Enz] in
the
vs. WT-GHvs. WT-GHmodel [
µ
M]vs. WT-GHvs. WT-GHmodel [
µ
M]vs. WT-GHvs. WT-GHmodel [
µ
M]vs. WT-GHvs. WT-GHmodel [
µ
M]
DXS
#0.580.830.024930.720.900.026890.780.920.027560.790.920.02768
DXR0.690.880.019860.410.720.016280.460.760.017080.410.730.01632
CMK0.700.890.019941.061.010.022741.871.180.026600.670.870.01960
HDS0.420.730.365570.850.940.472151.021.000.499881.141.030.51627
LS0.230.550.009380.400.720.012220.270.600.010220.440.750.01274
L3H0.350.670.002021.681.150.003450.140.400.001210.780.920.00275
PR6.861.570.0023610.881.710.002572.391.260.001881.261.060.00160
MFS0.140.400.000031.331.080.000080.170.460.000030.170.460.00003
* Enzyme concentrations of DXS, LS, L3H, PR and MFS were
determined experimentally based on Western analyses.
# Gene/enzyme identifiers as in Fig. 1 of manuscript
^ Abbreviations and acronyms: WT, wild-type; GH,
greenhouse-grown; LL, grown under low light conditions in growth
chamber; LW, grown under drought conditions in greenhouse;
LL/HT, grown under low light and high night temperature
conditions in growth chambers; MFS7, transgenic lines with reduced
(+)-menthofuran synthase gene expression levels;
Z, factor to account for (+)-menthofuran retained in secretory
cells; W, factor to account for (+)-pulegone retained in secretory
cells; GN(max), number of glandular trichomes at 30 d;
S0, total amount of available substrate for monoterpene
biosynthesis (calculated based on experimentally determined
essential oil yields).
$ The value used for the simulation in Fig. 5 of the main
manuscript corresponds to an estimated amount of 7,500 glanudlar
trichomes at 30 d. The (more accurate) experimentally
determined value is 8,262 glandular trichomes at 30 d.
WT - LL/HT
Z=1500; W=0.05; GN(max)=5017; S0=850
WT - LW
Z=400; W=0.05; GN(max)=7273; S0=1200
-5
5
15
25
35
0
10
20
30
40
50
60
c
a
b
-5
5
15
25
35
0
10
20
30
40
50
60
c
a
b
-5
5
15
25
35
0
10
20
30
40
50
60
c
a
b
The Gaussian function represents the following variables:
f(t) = a x exp (-(t-b)2)/2c2)
where
a = Concentration of enzyme in glandular trichomes [µM]
(parameter in the model)
t = Time after leaf emergence [s] (variable in the model)
b = Factor defining the the center of the Gaussian peak for
enzyme activity [s] (parameter in the model)
c = Factor defining the width of the Gaussian peak for enzyme
activity at half maximum [s] (parameter in the model)
C) Example of the use of two Gauss functions to approximate
enzyme activity data
To approximate the curve for developmental patterns of enzyme
activities with non-Gaussian shapes we used more than one Gaussian
function. The pattern of (+)-pulegone reductase activity will serve
as an example (plot of Gaussian graph with gray dotted lines):
The Gaussian function was further modified to account for the
volume density of the subcellular compartment in which each enzyme
resides (Table 3). This factor (“Comp” in the model) was taken
directly from Table 1 (e.g., leucoplasts account for roughly 13.9 %
of the area of an actively oil-secreting secretory cell, which
means that the “Comp” factor in the model would be 0.139). The
notation of the Gaussian function in the model is thus:
f(t) = Comp x a x exp (-(t-b)2)/2c2)
Table 3. Subcellular localization of enzymes involved in
peppermint monoterpene biosynthesis.
_______________________________________________________________________________________
EnzymeCompartment Reference
_______________________________________________________________________________________
1-Deoxy-D-xylulose 5-phosphate synthaseLeucoplastsLange et al.
(1998) Proc. Natl. Acad. Sci. USA 95: 2100
1-Deoxy-D-xylulose 5-phosphate reductoisomeraseLeucoplastsLange
et al. (1999) Arch. Biochem. Biophys. 365: 170
2C-Methyl-D-erythritol 4-phosphate
cytidyltransferaseLeucoplastsRohdich et al. (2000) Proc. Natl.
Acad.Sci. USA 97: 6451
4-(Cytidine 5’-diphospho)-2C-methyl-D-erythritolLeucoplastsLange
et al. (1999) Proc. Natl. Acad. Sci. USA 96: 13714
4-phosphate kinaseRohdich et al. (2000) Proc. Natl. Acad.Sci.
USA 97: 8251
2C-Methyl-D-erythritol 2,4-cyclodiphosphate
synthaseLeucoplastsGao et al. (2006) J. Biochem. Mol. Biol. 39:
502
(E)-4-Hydroxy-3-methyl-but-2-enyl diphosphate
synthaseLeucoplastsQuerol et al. (2002) FEBS Lett. 514: 343
(E)-4-Hydroxy-3-methyl-but-2-enyl diphosphate
reductaseLeucoplastsHsieh et al. (2005) Plant Physiol. 138: 641
Isopentenyl diphosphate isomeraseLeucoplastsTurner and Croteau
(2004) Plant Physiol. 136: 4215
(-)-Limonene synthase
LeucoplastsTurner et al. (1999) Plant Physiol. 120: 879
(-)-Limonene 3-hydroxylaseERTurner and Croteau (2004) Plant
Physiol. 136: 4215
(-)-trans-Isopiperitenol dehydrogenase
MitochondriaTurner and Croteau (2004) Plant Physiol. 136:
4215
(-)-Isopiperitenone reductaseCytosolTurner and Croteau,
unpublished data
(+)-Menthofuran synthaseERBertea et al. (2001) Arch. Biochem.
Biophys. 390: 279
(+)-Pulegone reductase CytosolTurner and Croteau (2004) Plant
Physiol. 136: 4215
(-)-Menthone:(-)-menthol reductaseCytosolTurner and Croteau,
unpublished data
(-)-Menthone:(+)-neomenthol reductaseCytosolTurner and Croteau,
unpublished data
_______________________________________________________________________________________
2.Estimating developmental dynamics of glandular trichome
density
Glandular trichome density and distribution depend on the
developmental status of the leaf under investigation, which has
been studied in detail by Rodney Croteau’s laboratory (Turner et
al. (2000) Plant Physiol. 124: 655-663). We expanded the studies to
investigate glandular trichome distributions on leaves of plants
grown under various environmental conditions (Scheme 2).
Scheme 2. Distribution of glandular trichomes on levaes of
wild-type plants grown under greenhouse conditions. The blue line
graph indicates the experimentally determined number of glandular
trichomes for each leaf size class (± standard error). The broken
black line depicts the logistic function used to approximate
glandular trichome numbers (for details see text).
0200040006000800010000010203040Glandular trichome countDays
after leaf emergence
To account for developmental dynamics in trichome density, we
introduced a logistic function, the most common sigmoid curve, to
approximate the curve shown in Scheme 2. This function specifies an
initial lag phase, then grows steeply but, because of limits in the
size of the glandular trichome population, eventually levels off at
the later stages of leaf development. Based on more recent data,
there is no drop in the total number of glandular trichomes at
later stages of development, when plants are grown under greenhouse
conditions (indicated by the red dotted line in Scheme 2). Thus, a
logistic function is an appropriate approach to approximate this
curve. The function was selected based on manual trial and error
experiments.
The expression for the logistic function for the number of
glandular trichomes (GN) as a function of time after leaf emergence
is:
GN(t) = a / (1 + c x expkt)
where
t = Time after leaf emergence [s] (variable in the model)
a = Number of glandular trichomes on a fully expanded leaf
(parameter in the model)
c = Number of times the initial gland population must grow to
reach “a” (parameter in the model)
k = Factor determining the slope during the growth phase of the
curve (parameter in the model)
The values used for the parameters a, c, and k in the model are
listed in Table 4.
Table 4. Parameters to approximate dynamics of glandular
trichome formation using a logistic function.
Experiment
Parameter a
Parameter c
Parameter k
WT-GH (wild-type grown under greenhouse conditions)
1
8 x 104
1.11 x 10-5
WT-LW (wild-type grown under low water conditions)
0.7
8 x 104
1.11 x 10-5
WT-LL (wild-type grown under low light conditions)
0.73
8 x 104
1.11 x 10-5
WT-LL/HT (wild type treated with combination of low water and
high night temperature conditions)
0.53
8 x 104
1.11 x 10-5
MFS7a-GH (MFS7a line grown under greenhouse conditions)
1.15
8 x 104
1.11 x 10-5
MFS7a-LL (MFS7a lines grown under low light conditions)
0.8262
8 x 104
1.11 x 10-5
L3H20-GH (L3H20 line grown under greenhouse conditions; Model
1)
1
8 x 104
1.11 x 10-5
L3H-GH (L3H20 line grown under greenhouse conditions; Model 2;
approximation of delayed glandular trichome formation)
1.09*
8 x 104*
1.11 x 10-5*
* For this experiment GN was set to 0.21 (experimentally
observed lag phase) for 0-15 d, and the logistic function (with
above listed parameters) was used to approximate GN from 15-40
d.
3.Kinetic properties of enzymes involved in peppermint
monoterpene biosynthesis
The laboratory of Rodney Croteau at Washington State University
determined the kinetic properties of all enzymes involved in the
peppermint p-menthane monoterpene biosynthetic pathway. Values
listed in Table 1 reflect data obtained using in vitro assays with
either purified recombinant or purified native enzymes. Values for
the enzymes involved in the methylerythritol pathway, which
provides the precursors for monoterpene biosynthesis, were taken
from the literature. Some of the kinetic constants had to be
estimated as no relevant information could be obtained directly
from the literature. The following rationales were use in these
parameter estimations:
(-)-Limonene 3-hydroxylase - Kcat value estimated based on
published data for other terpene hydroxylases, including
premnaspirodiene oxygenase.
(+)-cis-Isopulegone isomerase – the mechanism of enzyme
resembles that of ketosteroid isomerase; the Kcat value of this
enzyme ranges between 1.4 and 3.8; we used an average value of
Kcat=2.5 for simulations.
(+)-Menthofuran synthase - Km and Kcat values taken from
published work on psoralen synthase, a cytochrome P450-dependent
monooxygenase with the highest known sequence identity to
(+)-menthofuran synthase.
Table 5. Kinetic properties of enzymes involved in peppermint
monoterpene biosynthesis (numbering of enzymes as in Fig. 1 of main
manuscript).
_______________________________________________________________________________________
Enzyme
Km Kcat References
[mM] [s-1]
_______________________________________________________________________________________
(1) 1-Deoxy-D-xylulose 5-phosphate synthase (GAP) 0.068
1.9Eubanks and Poulter (2003) Biochemistry 42: 1140-1149
(Pyruvate) 0.44 1.9Eubanks and Poulter (2003) Biochemistry 42:
1140-1149
(2) 1-Deoxy-D-xylulose 5-phosphate (DXP) 0.132 4.4Rohdich et al.
(2006) FEBS J. 273: 4446-4458
reductoisomerase
(MEP) 0.972 1.6Rohdich et al. (2006) FEBS J. 273: 4446-4458
(3) 2C-Methyl-D-erythritol 4-phosphate
0.5 26Rohdich et al. (2000) PNAS 97: 6451-6456
cytidyltransferase
(4) 4-(Cytidine 5’-diphospho)-2C-methyl-D- 0.1 1Bernal et al.
(2005) Anal. Biochem. 250: 245-251.
erythritol 4-phosphate kinase
(5) 2C-Methyl-D-erythritol 2,4-cyclodiphosphate 0.252 3.4Rohdich
et al. (2001) Eur. J. Biochem. 268: 3190-3197
synthase
Shi et al. (2007) Biochem. Mol. Biol. 40: 911-920
(6) (E)-4-Hydroxy-3-methyl-but-2-enyl
0.42 0.4Kollas et al. (2002) FEBS Lett. 532: 432-436
diphosphate synthase
(7) (E)-4-Hydroxy-3-methyl-but-2-enyl
0.03 3.7Altincicek et al. (2002) FEBS Lett. 532: 437-440
diphosphate reductase
Graewert et al. (2004) J.Am.Chem.Soc. 126: 12847-12855
(8) Isopentenyl diphosphate isomerase (DMAPP) 0.0051
0.018Ramos-Valdivia et al. (1997) Eur.J.Biochem. 249: 161-170
(IPP) 0.017 0.89Ramos-Valdivia et al. (1997) Eur.J.Biochem. 249:
161-170
(9) Geranyl diphosphate synthase
(DMAPP) 0.054 48Burke et al. (1999) PNAS 96: 13062-13067
(IPP) 0.026 48Burke et al. (1999) PNAS 96: 13062-13067
(10) (-)-Limonene synthase
0.020 0.3Alonso et al. (1992) J. Biol. Chem. 267: 7582-7587
(11) (-)-Limonene 3-hydroxylase
0.018 1.5*Karp et al. (1990) Arch. Biochem. Biophys. 276:
219-226
Takahashi et al. (2007) J. Biol. Chem. 282: 31744-31754
(12) (-)-trans-Isopiperitenol dehydrogenase
0.072 0.002Ringer et al. (2005) Plant Physiol. 137: 863-872
(13) (-)-Isopiperitenone reductase
0.001 1.3Ringer et al. (2003) Arch. Biochem. Biophys. 4186:
80-92
(14) (+)-cis-Isopulegone isomerase
0.27 2.5*Kjonaas et al. (1985) Arch. Biochem. Biophys. 238:
49-60
(15) (+)-Menthofuran synthase
0.03* 2.0*Larbat et al. (2007) J. Biol. Chem. 282: 542-554.
(16) (+)-Pulegone reductase
0.0023 1.8Ringer et al. (2003) Arch. Biochem. Biophys. 4186:
80-92
(17) (-)-Menthone:(-)-menthol reductase (menthone) 0.003
0.6Davis et al. (2005) Plant Physiol. 137: 873-881
((+)-isomenthone) 0.041 0.6Davis et al. (2005) Plant Physiol.
137: 873-881
(18) (-)-Menthone:(+)-neomenthol reductase (menthone) 0.674
0.06Davis et al. (2005) Plant Physiol. 137: 873-881
((+)-isomenthone) 1.0 0.06Davis et al. (2005) Plant Physiol.
137: 873-881
_________________________________________________________________________________________________________
* These values could not be obtained from the literature and
have thus been estimated.
4.Generating a system of ordinary differential equations to
describe kinetic properties of enzymes
The Michaelis-Menten rate equation, as developed by Briggs and
Haldane (Fersht (1985) In: Enzyme structure and mechanism, Ed 2.
New York: W.H. Freeman), allows calculating the change of the
concentration of a metabolite based on the rate of enzymatic
formation and turnover. Using the monoterpene pathway intermediate
(-)-trans-isopiperitenol as an example we obtain the following:
where
Kc11, Kcat(limonene 3-hydroxylase); E11, Concentration(limonene
3-hydroxylase); Kc12, Kcat(trans-isopiperitenol dehydrogenase);
IPPol, (-)-trans-isopiperitenol; LM, (-)-limonene; KM11,
Km(limonene 3-hydroxylase); KM12, Km(trans-isopiperitenol
dehydrogenase).
Three enzymes in the monoterpene pathway catalyze fully
reversible reactions: 1-deoxy-D-xylulose 5-phosphate
reductoisomerase (DXR), isopentenyl diphosphate isomerase and
(-)-menthone:(+)-neomenthol reductase. The formalism for these
reactions is different. For example, the time-dependent change in
the concentration of 1-deoxy-D-xylulose 5-phosphate is calculated
as follows:
Formation
Turnover
where
(for turnover of 1-deoxy-D-xylulose 5-phosphate by DXR)
Kc2f, Kcat(1-deoxy-D-xyxlulose 5-phosphate reductoisomerase;
forward reaction); Kc2r, Kcat(1-deoxy-D-xyxlulose 5-phosphate
reductoisomerase; reverse reaction); KM2f, Km(1-deoxy-D-xyxlulose
5-phosphate reductoisomerase; forward reaction); KM2r,
Km(1-deoxy-D-xyxlulose 5-phosphate reductoisomerase; reverse
reaction); E2, Concentration(1-deoxy-D-xyxlulose 5-phosphate
reductoisomerase); DOXP, 1-deoxy-D-xylulose 5-phosphate; ME4P,
2C-methyl-D-erythritol 4-phosphate.
The reaction catalyzed by (+)-pulegone reductase yields
(-)-menthone and (+)-isomenthone in a 10 : 1 ratio. We used two
separate expressions for these reactions (basically treating the
two reactions as being catalyzed by two different enzymes).
The enzyme (-)-menthone:(-)-menthol reductase accepts two
substrates ((-)-menthone and (+)-isomenthone) and converts them
into two different products ((-)-menthol and (+)-neoisomenthol).
Since the mechanism of this reaction is unknown, these two
reactions are treated as being catalyzed by two different enzymes.
The same is true for the enzyme (-)-menthone:(+)-neomenthol
reductase (substrates: (-)-menthone and (+)-isomenthone; products:
(+)-neomenthol and (+)-isomenthol).
Two enzymes involved in peppermint monoterpene biosynthesis are
known to be affected by competitive feedback inhibition:
isopentenyl diphosphate isomerase and (+)-pulegone reductase. As an
example, modified Michaelis-Menten rate equations are used to
account for the effect of the competitive inhibitor (+)-menthofuran
on the turnover of (+)-pulegone by (+)-pulegone reductase. In
addition to assessing the inhibition of (+)-pulegone reductase by
(+)-menthofuran, we also considered substrate inhibition (as
determined experimentally):
where
Kc14, Kcat((+)-isopulegone isomerase); E14,
Concentration((+)-isopulegone isomerase); KM14, Km((+)-isopulegone
isomerase); CIPUL, (+)-cis-isopulegone; Kc15, Kcat((+)-menthofuran
synthase); E15, Concentration((+)-menthofuran synthase); KM15,
Km((+)-menthofuran synthase); MF, (+)-menthofuran; Kc16a,
Kcat((+)-pulegone reductase; (-)-menthone-forming); E16a,
Concentration((+)-pulegone reductase; (-)-menthone-forming); KM16a,
Km((+)-pulegone reductase; (-)-menthone-forming); PUL,(+)-pulegone;
Kc16b, Kcat((+)-pulegone reductase; (+)-isomenthone-forming); E16b,
Concentration((+)-pulegone reductase; (+)-isomenthone-forming);
KM16b, Km((+)-pulegone reductase; (+)-isomenthone-forming); Kis,
substrate inhibition constant of (+)-pulegone on (+)-pulegone
reductase; Kic, feedback competitive inhibition constant for
(+)-menthofuran on (+)-pulegone reductase.
The above expression (d[PUL]/dt) also contains two additional
factors “w” and “z”, which account for the fact that (+)-pulegone
reductase (PR) can be affected by substrate inhibition (w-factor)
and competitive inhibition by the pathway side product
(+)-menthofuran. In a recent publication (Rios-Estepa et al. (2008)
Proc. Natl. Acad. Sci. USA 105: 2818-2823) we reported that the
concentration of (+)-menthofuran was roughly 400 µM in secretory
cells obtained from plants grown under greenhouse conditions. Based
on the experimentally determined monoterpene profiles of these
plants, inhibitiory effects on PR were negligible. The z-factor is
used to modify the actual (+)-menthofuran concentration in
secretory cells. Our measurements indicated that the concentration
of this compound in secretory cells (which is where PR is present)
is 100 times less than in the essential oil that accumulates
extracellularly in the subcuticular cavity. The expression for
competitive inhibition of PR contains the (+)-menthofuran
concentration in the denominator (see above). A z-factor of 100
(which is used for greenhouse conditions) thus reduces the effect
of competitive inhibition to reflect our experimental data. Under
low light conditions, the total concentration of (+)-menthofuran in
secretory cells was determined as 20 mM, with an estimated
concentration of 6 mM in the cytosol. The latter value corresponds
to the solubility product of (+)-menthofuran in aqueous solutions
(Fichan et al. (1999) J. Chem. Eng. Data 44: 56-62). Based on
electron microscopic images the remaining (+)-menthofuran appears
to partition into membranes (data not shown). Based on these
experimental measurements, the (+)-menthofuran concentration in the
cytosol of secreotry cells (which is the location of (+)-pulegone
reductase) is 15-fold higher when plants are grown under low light
conditions compared to greenhouse-grown plants. This is reflected
in a 15-fold higher z-factor value (1,500 for plants grown under
low light). The concentration of (+)-menthofuran in secretory cells
of plants grown under low water conditions is also higher than in
greenhouse-grown plants but the increase is only 4-fold (which is
reflected in a 4-fold higher z-factor). The MFS7a mutant line
accumulates (+)-menthofuran at only very low levels (10-fold less
than wild-type grown under greenhouse conditions) and the z-factor
was thus adjusted to 10. In stress-treated conditions MFS7a plants
(+)-menthofuran is accumulated to higher levels than in
green-house-grown plants (20-fold higher under low light and
40-fold higher under low light; reflected in a 20- and 40-fold
higher z-value, respectively) but remains significantly below the
corresponding levels in wild-type plants. The factor “w” is used in
an analogous fashion to account for the actual concentration of
(+)-pulegone in secretory cells. Based on experimental data
(Rios-Estepa et al. (2008) Proc. Natl. Acad. Sci. USA 105:
2818-2823) the concentration of (+)-pulegone in secretory cells is
5 % of the total concentration in glandular trichomes and we thus
use a w-factor of 0.05 to adjust the (+)-pulegone concentration in
the expression for substrate inhibition. It was also observed
experimentally that the (+)-pulegone concentration in secretory
cells was only marginally affected by environmental conditions and
we therefore use a w-factor of 0.05 for all simulations.
5.Calculating monoterpenoid essential oil yields for individual
glandular trichomes
Our model simulates monoterpene composition and yield for
individual trichomes, which is then extrapolated to the entire leaf
using a logistic function that accounts for the number and
developmental distribution on glandular trichomes (see 2. for
details). The key constraints for our model are experimentally
determined monoterpene levels. It is thus essential to be able to
extrapolate from these macroscopic measurements (essential oil
distillations from entire leaves) to the scale of an individual
trichome. In other words, we needed to determine the storage
capacity of a glandular trichome.
The volume of the essential oil-filled subcuticular cavity of
mature glandular trichomes (Scheme 1, Lower Panel) was originally
calculated based on the approximation of its shape as a hemisphere
(2/3 π r3). However, we noticed that, using this approach, the
amount of oil was significantly underestimated. We thus modified
our estimation of the oil storage cavity by approximating it as a
sphere (volume: 4/3 π r3) minus the volume of the secretory cells
(see 1.1 for details). Trichomes were divided into three different
classes: large (75-82 μm diameter; average radius 39 μm), medium
(65-74 μm diameter; average radius 35 μm) and small (50-64 μm
diameter; average radius 30 μm). The volumes were thus calculated
as 2.25 x 10-4 µl (large-sized trichomes), 1.56 x 10-4 µl
(medium-sized trichomes) or 0.73 x 10-4 µl (small-sized trichomes).
Using this calculation the amount of oil was slightly
overestimated, potentially because of the presence of a previously
described non-oil mucilage layer surrounding the seceretory cells,
as indicated by black dots in Scheme 3 (Turner et al. (2000) Plant
Physiol. 124: 665-680). This was corrected by introducing a factor
of 0.94, which led to accurate representations of oil volumes (2.03
x 10-4 µl for large-sized trichomes, 1.40 x 10-4 µl for
medium-sized trichomes) or 0.66 x 10-4 µl for small-sized
trichomes).
Scheme 3
0
10
20
30
0102030405060
Enzyme activity
[
µ
mol/h/leaf]
Leaf age [d]
(+)-cis-Isopulegoneisomerase
Y (-)-Isopiperitenonereductase
X (-)-trans-Isopiperitenoldehydrogenase
+(+)-Pulegonereductase
-(-)-Menthonereductase
▲(-)-Limonene 3-hydroxylase
■(-)-Limonene synthase
0
10
20
30
0102030405060
Enzyme activity
[
µ
mol/h/leaf]
Leaf age [d]
(+)-cis-Isopulegoneisomerase
Y (-)-Isopiperitenonereductase
X (-)-trans-Isopiperitenoldehydrogenase
+(+)-Pulegonereductase
-(-)-Menthonereductase
▲(-)-Limonene 3-hydroxylase
■(-)-Limonene synthase
For example, leaves of peppermint plants grown under greenhouse
conditions contained 39 % large, 57 percent medium and 4 % small
glandular trichomes at 30 d after leaf emergence. A total of 10151
glandular trichomes was counted, thus indicating a distribution of
3,959 large (39 %), 5786 medium (57 %), and 406 small (4 %)
glandular trichomes. In order to calculate the amount of oil per
leaf, the number of glandular trichomes in each size category was
multiplied by the appropriate trichome volume (e.g, 3,959 (number
of large trichomes per leaf) x 2.08 x 10-4 µl (volume of individual
large trichome) = 0.822 µl per leaf). The known density of
peppermint essential oil (0.9) then allows us to estimate essential
oil yield (in µg per leaf). These types of calculations are the
basis of all tables shown in the main manuscript.
6.Estimating changes in enzyme concentrations based on
measurements of gene expression levels
The concentrations of enzymes involved in peppermint monoterpene
biosynthesis in glandular trichomes were determined for greenhouse
growth conditions as described in chapter 1. However, it is not
practically feasible to determine enzyme concentrations under
various environmental conditions, at different stages of leaf
development, and in different transgenic plants. Thus, we
approximated differences between greenhouse-grown wild-type plants
and experimental plants by acquiring quantitative real-time PCR
data regarding the expression of key biosynthetic genes and
subsequently extrapolated differences in gene expression levels to
changes in enzyme concentrations. This is possible because the
Lange and Croteau laboratories at WSU have accumulated a wealth of
experimental data regarding the correlation of gene expression
levels and enzyme activities (for 5 gene/enzyme pairs) in wild-type
peppermint plants at many different developmental stages (Scheme
4).
Scheme 4. Correlation of gene expression and enzyme activity
patterns in peppermint leaf glandular trichomes. The x-axis
indicates gene expression changes, whereas the y-axis shows enzyme
activity changes as fold-change from the value measured at 15 d
after leaf emergence.
y = 0.3013ln(x) + 0.9937R² = 0.9529
0.00.51.01.52.00.02.04.06.0
The gene expression and enzyme activity levels measured at 12 d
after leaf emergence were set to “1”. All other gene
expression/enzyme activity pairs were calculated as a fold-change
from this calibrator value. As indicated by an analysis of 32
measured gene expression/enzyme activity pairs throughout leaf
development (8 to 40 d after leaf emergence) (Scheme 4), there is a
correlation between the change in the levels of a certain
transcript and the change in the corresponding enzyme activity.
This correlation can be described by the equation y = 0.3013 ln (x)
+ 0.9937, which is a logarithmic function (R2 = 0.9529). We assume
that the same equation can be used to approximate enzyme
concentrations from gene expression levels (Table 6).
It is known from various published experiments that, under
regular greenhouse conditions, peppermint essential oil
biosynthesis is regulated primarily at the transcriptional level
(Gershenzon et al. (2000) Plant Physiol. 122: 205-214; McConkey et
al. (2000) Plant Physiol. 122: 215-224). One of the exceptions to
this rule was identified in our previous work, when we identified
the importance of feedback control by the dead-end pathway side
product (+)-menthofuran) on (+)-pulegone reductase activity under
low light conditions (Rios-Estepa et al. (2008) Proc. Natl. Acad.
Sci. USA 105: 2818-2823). In the current study we used the same
approach to investigate if additional as yet unknown regulatory
processes might need to be considered (which would be the case when
modeling and experimental data match poorly). It is important to
note that we did not perform any modeling optimizations in the
present work. The primary focus of this study was to evaluate the
minimum set of experimental data that, when incorporated into our
model, generate simulations reflecting experimentally determined
monoterpene profiles.
7.Kinetic model for peppermint monoterpene biosynthesis
Our recently published first generation model (Rios-Estepa et
al. (2008) Proc. Natl. Acad. Sci. USA 105: 2818-2823) encompassed
the core reactions of the p-menthane pathway of monoterpene
biosynthesis in peppermint glandular trichomes. The current work
builds on this model and extends it to include additional reactions
(precursor supply in leucoplasts of glandular trichome cells). Our
modeling applies the conservation law of mass to secretory cell as
the reaction volume. We did not perform parameter optimizations as
kinetic and other parameters were inferred directly from
experimental data. Statistical tests (primarily the Chi Square
test) were used to evaluate the goodness of fit of simulated versus
experimentally determined monoterpene profiles. We are currently
not considering transport processes or thermodynamics.
The variation of a metabolite M over time is proportional to the
difference between the rate at which it is formed (anabolic
reaction, RA) minus the rate at which it is turned over (catabolic
reaction, RC). The model combines the mass balances for each
individual metabolite of the peppermint monoterpene pathway into a
series of stiff ordinary differently equations (ODEs), which must
be solved simultaneously:
dM/dt = RA – RC
The metabolite-forming reaction (assuming Michaelis - Menten
type kinetics) is defined as:
RA= [EA] kcat) [M]/(KM + [M]),
Our model assumes a limited supply of precursors for monoterpene
biosynthesis, pyruvate and glyceraldehydes 3-phosphate ([S0] =
([Pyr] + [GAP]), which is calculated based on the final amount of
oil produced by glandular trichomes under various conditions (oil
yields are determined experimentally). One option for solving a
system of ordinary differential equations in the MATLAB framework
is the ode45 solver, which uses the Runge Kutta Higher order
method. However, this method does not work well with stiff
differential equations (Harman et al. (2000) Advanced Engineering
Mathematics with MATLAB®, 2nd Ed., Cengage Learning, Florence, KY).
In such cases, the ode15s solver is recommended
(http://www.mathworks.com/access/helpdesk/help/pdf_doc/otherdocs/ode_suite.pdf).
For the numerical solution of the stiff ODE system, we wrote a
MATLAB program that contains the following files, functions,
parameters and variables:
A) Script File
· A set of commands that includes the vector for pathway
metabolites, time span, and the vector of initial conditions.
· It calls the function (m-file) that solves the ODEs and
produces the graphical outputs (monoterpene profiles).
B) Function ( m-file)
· Inputs: independent variable t (time span); vector of
dependent variables x ([Metabolites]).
· Solves the set of ODEs with the initial values given in the
vector of initial conditions. Returns the values of the independent
variable in the vector t (time span) and the values of the
dependent variables in the vector x ([Metabolites]). The vector of
independent variables t is not equally spaced because the function
(m-file) controls the step size.
C) Parameters
· Kinetic constants of enzymes involved in p-menthane
monoterpene biosynthesis. Not optimized because these values were
inferred directly from experimental data.
· w-Factor accounts for the small amounts of (+)-pulegone
retained in secretory cells (does not change under various
environmental conditions). Not optimized because this value was
inferred directly from experimental data.
· Reaction volume: volume of secretory cells of glandular
trichomes, which was inferred directly from experimental data.
D) Non-constant parameters (variables)
· Independent variable t (time span); dependent variable x
([Metabolites])
· Gauss function to approximate dynamic changes in enzyme
concentrations over time (d[E]/dt = f( a1-a18, b1-b18, c1-c18)).
Not optimized because the values for parameters a, b and c were
inferred directly from experimental data.
· Logistic function to approximate dynamic changes in the
distribution of leaf glandular trichomes over time (GN = f(a, c,
k)). Not optimized because the values for parameters a, c and k
were inferred directly from experimental data.
· z-Factor accounts for the selective retention of
(+)-menthofuran in secretory cells under stress conditions (z =
f(phenotype, environmental conditions)). Not optimized because this
value was inferred directly from experimental data.
7.1Reaction mechanisms utilized in kinetic model of peppermint
monoterpene biosynthesis
A mechanism following regular Michaelis-Menten-type kinetics is
assumed for all
enzymes with the following exceptions:
(1) Substrate inhibition of (+)-pulegone reductase
(2) Competitive inhibition of (+)-pulegone reductase by
(+)-menthofuran
(3) Competitive inhibition of isopentenyl-diphosphate isomerase
by GPP
(4) Reversible reaction mechanisms were assumed for
1-Deoxy-D-xylulose-5-
phosphate reductoisomerase and (-)-Menthone:(+)-neomenthol
reductase
(5) Bi-bi (two substrates, two products) reaction mechanisms
were assumed for
1-deoxy-D-xylulose-5-phosphate synthase (Pyruvate + GAP = DXP +
CO2) and
geranyl diphosphate synthase (IPP + DMAPP = GPP + PPi). The
former utilizes
an ordered mechanism (Pyr binds first), whereas a random
mechanism is assumed
for the latter.
Metabolites
GAP: Glyceraldehyde-3-phosphate
Pyr: Pyruvate
DOXP: 1-Deoxy-D-xylulose-5-phosphate
MEP4: 2-C-Methyl-D-erythritol 4-phosphate
CDPME: 4-(Cytidine 5'-diphospho)-2-C-methyl-D-erythritol
CDPME2P: 2-Phospho-4-(Cytidine
5'-diphospho)-2-C-methyl-D-erythritol
MecPP: 2-C-Methyl-D-erythritol 2,4-cyclodiphosphate
HMBPP: 4-Hydroxy-3-methylbut-2-en-1-yl diphosphate
DMAPP: Dimethylallyl-pyrophosphate
IPP: Isopentenyl-pyrophosphate
GPP: Geranyl diphosphate
LM: (-)-Limonene
IPPol: (-)-trans-Isopiperitenol
IPPone: (-)-Isopiperitenone
CIPUL: (+)-Cis-Isopulegone
PUL: (+)-Pulegone
MF: (+)-Menthofuran
Imone: (+)-isomenthone
Mone: (-)-Menthone
Nmol: (+)-Neomenthol
Mol(: (-)-Menthol
Imol: (+)-Isomenthol
NIMol: (+)-Neoisomenthol
Species equations
Variation of GAP
Variation of Pyr
Variation of DOXP
Variation of ME4P
Variation of CDPME
Variation of CDPME2P
Variation of MEcPP
Variation of HMBPP
Variation of DMAPP
Variation of IPP
Variation of GPP
Variation of LM
Variation of IPPol
Variation of IPPone
Variation of CIPUL
Variation of PUL
Variation of MF
Variation of IMone
Variation of Mone
Variation of NMol
Variation of Mol
Variation of IMol
Variation of NIMol
7.2Code of reference model for wild-type plants grown under
greenhouse conditions
function xdot = mint_MEP6_GH_WT(t,x)
% This function calculates monoterpene amounts (40 day time
course) in leaves
% of peppermint WT plants grown in a greenhouse with
supplemental lighting from sodium
% vapor lights.
% A mechanism following regular Michaelis-Menten-type kinetics
is assumed for all
% enzymes with the following exceptions:
% (1) Substrate inhibition of (+)-pulegone reductase
% (2) Competitive inhibition of (+)-pulegone reductase by
(+)-menthofuran
% (3) Competitive inhibition of isopentenyl-diphosphate
isomerase by GPP
% (4) Reversible reaction mechanisms were assume for
1-Deoxy-D-xylulose-5-phosphate
% reductoisomerase and (-)-Menthone:(+)-neomenthol reductase
% (5) Bi-bi (two substrates, two products) reaction mechanisms
were assumed for
% 1-deoxy-D-xylulose-5-phosphate synthase (Pyruvate + GAP = DXP
+ Co2) and geranyl
% diphosphate synthase (IPP + DMAPP = GPP + PPi). The former
utilizes an ordered mechanism
% (Pyr binds first), whereas a random mechanism is assumed for
the latter.
% Metabolite Nomenclature
%[GAP]=x(1) D-Glyceraldehyde 3-Phosphate
%[Pyr]=x(2) Pyruvate
%[DOXP]=x(3) 1-Deoxy-D-xylulose 5-phosphate
%[ME4P]=x(4) 2-C-Methyl-D-erythritol-4-phosphate
%[CDPME]=x(5) 4-(Cytidine
5'-diphospho)-2-C-methyl-D-erythritol
%[CDPME2P]=x(6) 2-Phospho-4-(cytidine
5'-diphospho)-2-C-methyl-D-erythritol
%[MEcPP]=x(7) 2-C-Methyl-D-erythritol-2,4-cyclodiphosphate
%[HMBPP]=x(8) 1-Hydroxy-2-methyl-2-(E)-butenyl 4-diphosphate
%[DMAPP]=x(9) Dimethylallyl-pyrophosphate
%[IPP]=x(10) Isopentenyl diphosphate
%[GPP]=x(11) Geranyl diphosphate
%[LIM]=x(12) (-)-Limonene
%[IPPol]=x(13) (-)-trans-Isopiperitenol
%[IPPone]=x(14) (-)-Isopiperitenone
%[CIPUL]=x(15) (+)-cis-Isopulegone
%[PUL]=x(16) (+)-Pulegone
%[MF]=x(17) (+)-Menthofuran
%[IMone]=x(18) (+)-Isomenthone
%[Mone]=x(19) (-)-Menthone
%[NMol]=x(20) (+)-Neomenthol
%[Mol]=x(21) (-)-Menthol
%[IMol]=x(22) (+)-Isomenthol
%[NIMol]=x(23) (+)-Neoisomenthol
% Kinetic Parameters
% kc units: [1/s] (kc = Kcat)
% KM units: [uM]
% Ki units: [uM]
KM1a = 68; %1-Deoxy-D-xylulose-5-phosphate synthase (DXS) for
GAP
kc1a = 1.9;
KM1b = 440; %1-Deoxy-D-xylulose-5-phosphate synthase (DXS) for
Pyr
kc1b = 1.9;
Kia = 16; %Dissociation constant for Pyr
KM2f = 132; %1-Deoxy-D-xylulose-5-phosphate reductoisomerase
(DXR; forward reaction)
kc2f = 4.4;
KM2r = 972; %1-Deoxy-D-xylulose-5-phosphate reductoisomerase
(DXR; reverse reaction)
kc2r = 1.6;
KM3 = 500; %2-C-Methyl-D-erythritol 4-phosphate
cytidylyltransferase (MCT)
kc3 = 26;
KM4 = 100; %4-(Cytidine 5'-diphospho)-2-C-methyl-D-erythritol
kinase (CMK)
kc4 = 1;
KM5 = 252; %2-C-Methyl-D-erythritol 2,4-cyclodiphosphate
synthase (MECPS)
kc5 = 3.4;
KM6 = 420; %4-Hydroxy-3-methylbut-2-en-1-yl diphosphate synthase
(HDS)
kc6 = 0.4;
KM7 = 30; %4-Hydroxy-3-methylbut-2-en-1-yl diphosphate reductase
(HDR)
kc7 = 3.7;
KM8f = 5.1; %Isopentenyl-diphosphate delta-isomerase for IPP
(IPPI; forward reaction)
kc8f =0.018;
KM8r = 17; %Isopentenyl-diphosphate delta-isomerase for DMAPP
(IPPI; rev reaction)
kc8r = 0.89;
KM9a = 54; %Geranyl diphosphate synthase (GPPS; DMAPP as
substrate)
kc9a = 48;
KM9b = 26; %Geranyl diphosphate synthase (GPPS; IPP as
substrate)
kc9b = 48;
KM10 = 20; %(-)-Limonene synthase (LS)
kc10 = 0.3;
KM11 = 18; %(-)-Limonene 3-hyroxylase (L3H)
kc11 = 1.8;
KM12 = 72; %(-)-trans-Isopiperitenol dehydrogenase (IsoDH)
kc12 = 0.002;
KM13 = 1; %(-)-Isopiperitenone reductase (IsoR)
kc13 = 1.3;
KM14 = 270; %(+)-cis-Isopulegone isomerase (IsoI)
kc14 = 2.5;
KM15 = 30; %(+)-Menthofuran synthase (MFS)
kc15 = 2.0;
KM16a = 2.3; %(+)-Pulegone reductase (PR; product:
(-)-menthone)
kc16a = 1.8;
KM16b = 2.3; %(+)-Pulegone reductase (PR; product:
(+)-isomenthone)
kc16b = 1.8;
KM17a = 3; %(-)-Menthone:(-)-menthol reductase (MMR; substrate:
(-)-menthone)
kc17a = 0.6;
KM17b = 41; %(-)-Menthone:(-)-menthol reductase (MMR; substrate:
(+)-isomenthone)
kc17b = 0.6;
KM18af = 674; %(-)-Menthone:(+)-neomenthol reductase (MNR;
substrate: (-)-menthone);
forward reaction)
kc18af = 0.06;
KM18ar = 1200; %(-)-Menthone:(+)-neomenthol reductase (MNR;
substrate: (-)-menthone);
backward reaction)
kc18ar = 0.06;% estimated
KM18b = 1000; %(-)-Menthone:(+)-neomenthol reductase (MNR;
substrate: (+)-isomenthone)
kc18b = 0.06;
Kic1=96; % Product inhibition constant (Geranyl diphosphate
acting on IPPI)
Kic2=300; % Product inhibition constant ((+)-menthofuran acting
on PR)
% Competitive inhibition mechanism
Kis=112; % Substrate Inhibition constant ((+)-pulegone acting on
PR)
% Uncompetitive inhibition mechanism
z=100; % Factor to account for the actual concentration of
(+)-menthofuran in
secretory cells of glandular trichomes
w=0.05; % Factor to account for the actual concentration of
(+)-pulegone in
secretory cells of glandular trichomes
% The model also takes into account that each enzyme shows a
particular transient
% pattern of expression. This pattern is approximated by a Gauss
function.
%First peak of activity:
%f(x) = Comp * a * exp((-(t-b).^2)/(2*(c)^2))
%where Comp = Factor to adjust for the volume density of the
compartment in which a
particular enzyme is active [Dimensionless]
% a = Factor defining the height of the Gaussian peak for enzyme
activity
[µM]
% t = Time [s]
% b = Factor defining the position of the center of the Gaussian
peak for
enzyme activity [s]
% c = Factor defining the width of the Gaussian peak for enzyme
activity at
half maximum [s]
b1=1296000; % Defines the position of the center of the Gaussian
peak for enzyme
activity.
% Relevant to the following enzyme activities: LS, L3H, IsoDH,
IsoR, IsoI,
MFS, PR
c1=800000; % Defines the width of the Gaussian peak for enzyme
activity at half
maximum.
% Relevant to the following enzyme activities: LS, L3H, IsoDH,
IsoR, IsoI,
MFS, PR
b5=1800000; % Defines the position of the center of the Gaussian
peak for enzyme
activity.
% Relevant to the following enzyme activities: MMR, MNR
c5=900000; % Defines the width of the Gaussian peak for enzyme
activity at half
maximum.
% Relevant to the following enzyme activities: MMR, MNR
E1=(0.139)*0.03*exp((-(t-b1).^2)/(2*(c1)^2)); % DXS
E2=(0.139)*0.0225*exp((-(t-b1).^2)/(2*(c1)^2)); % DXR
E3=(0.139)*0.5*exp((-(t-b1).^2)/(2*(c1)^2)); % MCT
E4=(0.139)*0.0225*exp((-(t-b1).^2)/(2*(c1)^2)); % CMK
E5=(0.139)*0.5*exp((-(t-b1).^2)/(2*(c1)^2)); % MECPS
E6=(0.139)*0.5*exp((-(t-b1).^2)/(2*(c1)^2)); % HDS
E7a=(0.139)*0.2*exp((-(t-b1).^2)/(2*(c1)^2)); % HDR (product:
DMAPP)
E7b=(0.139)*0.04*exp((-(t-b1).^2)/(2*(c1)^2)); % HDR (product:
IPP)
E8=(0.139)*0.3*exp((-(t-b1).^2)/(2*(c1)^2)); % IPPI
E9=(0.139)*0.1*exp((-(t-b1).^2)/(2*(c1)^2)); % GPPS
E10= (0.139)*0.017*exp((-(t-b1).^2)/(2*(c1)^2)); % LS
E11= (0.365)*0.003*exp((-(t-b1).^2)/(2*(c1)^2)); % L3H
E12= (0.044)*10*exp((-(t-b1).^2)/(2*(c1)^2)); % IsoDH
E13= (0.204)*0.34*exp((-(t-b1).^2)/((2*c1)^2)); % IsoR
E14= (0.204)*0.34*exp((-(t-b1).^2)/((2*c1)^2)); % IsoI
E15= (0.365)*0.00007*exp((-(t-b1).^2)/(2*(c1)^2)); % MFS
E16a=(0.204)*0.0015*exp((-(t-b1).^2)/(2*(c1)^2)); % PR (product:
(-)-menthone)
E16b=(0.204)*0.00015*exp((-(t-b1).^2)/(2*(c1)^2)); % PR
(product: (+)-isomenthone)
E17a=(0.204)*0.0011*exp((-(t-b5).^2)/(2*(c5)^2)); % MMR
(product: (-)-menthol)
E17b=(0.204)*0.0011*exp((-(t-b5).^2)/(2*(c5)^2)); % MMR
(product: (+)-neoisomenthol)
E18a=(0.204)*0.00001*exp((-(t-b5).^2)/(2*(c5)^2)); % MNR
(product: (+)-neomenthol)
E18b=(0.204)*0.00001*exp((-(t-b5).^2)/(2*(c5)^2)); % MNR
(product: (+)-isomenthol)
% The model also takes into account that the glandular trichome
density (GN) changes over time. This behavior is approximated using
a logistic function:
c=5*10^5; % parameter approximating slope of exponential phase
of sigmoid curve
k=1/8*10^4; % parameter approximating shape of sigmoid curve
GN = 1+ 1/(1+c*exp(-k*t)); % at day 15, gland number is 86.7 %
of total gland number at
day 30
%Species Equations
if t< 1296000 % (patterns of enzymes from 0 to 15 days after
leaf initiation)
xdot=[GN*(-(kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2))));
% Variation of GAP
GN*(-(kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2))));
% Variation of Pyruvate (same expression as for GAP)
GN*((kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2)))-((KM2r*kc2f*E2*x(3)-KM2f*kc2r*E2*x(4))/(KM2f*KM2r+KM2r*x(3)+KM2f*x(4))));
% Variation of DOXP
GN*(((KM2r*kc2f*E2*x(3)-KM2f*kc2r*E2*x(4))/(KM2f*KM2r+KM2r*x(3)+KM2f*x(4)))-(kc3*E3*x(4)/(x(4)+KM3)));
% Variation of ME4P
GN*((kc3*E3*x(4)/(x(4)+KM3))-(kc4*E4*x(5)/(x(5)+KM4))); %
Variation of CDP-ME
GN*((kc4*E4*x(5)/(x(5)+KM4))-(kc5*E5*x(6)/(x(6)+KM5))); %
Variation of CDP-ME2P
GN*((kc5*E5*x(6)/(x(6)+KM5))-(kc6*E6*x(7)/(x(7)+KM6))); %
Variation of MEcPP
GN*((kc6*E6*x(7)/(x(7)+KM6))-(kc7*E7a*x(8)/(x(8)+KM7))-
(kc7*E7b*x(8)/(x(8)+KM7))); % Variation of HMB-PP
GN*((kc7*E7a*x(8)/(x(8)+KM7))+(kc8f*E8*x(10)/(x(10)+KM8f*(1+(x(11)/Kic1))))-(kc8r*E8*x(9)/(x(9)+KM8r*(1+(x(11)/Kic1))))-((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b)));
%Variation of DMAPP
GN*((kc7*E7b*x(8)/(x(8)+KM7))+(kc8r*E8*x(9)/(x(9)+KM8r*(1+(x(11)/Kic1))))-(kc8f*E8*x(10)/(x(10)+KM8f*(1+(x(11)/Kic1))))-((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b)));
%Variation of IPP
GN*(((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b))-
(kc10*E10*x(11)/(x(11)+KM10))); %Variation of GPP
GN*((kc10*E10*x(11)/(x(11)+KM10))-(kc11*E11*x(12)/(x(12)+KM11))); %
Variation of LIM
GN*((kc11*E11*x(12)/(x(12)+KM11))-(kc12*E12*x(13)/(x(13)+KM12))); %
Variation of IPPol
GN*((kc12*E12*x(13)/(x(13)+KM12))-(kc13*E13*x(14)/(x(14)+KM13))); %
Variation of IPPone
GN*((kc13*E13*x(14)/(x(14)+KM13))-(kc14*E14*x(15)/(x(15)+KM14))); %
Variation of CIPUL
GN*((kc14*E14*x(15)/(x(15)+KM14))-(kc16a*E16a*x(16)/(x(16)+KM16a*(1+z*(x(17))/Kic2)))-(kc16b*E16b*x(16)/(x(16)+KM16b*(1+z*(x(17))/Kic2)))-(w*kc16a*E16a*x(16)/(KM16a+x(16)*(1+x(16)/Kis)))-(w*kc16b*E16b*x(16)/(KM16b+x(16)*(1+x(16)/Kis)))-(kc15*E15*x(16)/(x(16)+KM15)));
% Variation of PUL
GN*(kc15*E15*x(16)/(x(16)+KM15)); % Variation of MF
GN*((kc16b*E16b*x(16)/(x(16)+KM16b*(1+z*(x(17))/Kic2)))+(w*kc16b*E16b*x(16)/(KM16b+x(16)*(1+x(16)/Kis)))-(kc17b*E17b*x(18)/(x(18)+KM17b))-(kc18b*E18b*x(18)/(x(18)+KM18b)));
% Variation of IMone
GN*((kc16a*E16a*x(16)/(x(16)+KM16a*(1+z*(x(17))/Kic2)))+(w*kc16a*E16a*x(16)/(KM16a+x(16)*(1+x(16)/Kis)))-((KM18ar*kc18af*E18a*x(19)-KM18af*kc18ar*E18a*x(20))/(KM18af*KM18ar+KM18ar*x(19)+KM18af*x(20)))-(kc17a*E17a*x(19)/(x(19)+KM17a)));
% Variation of Mone
GN*((KM18ar*kc18af*E18a*x(19)-KM18af*kc18ar*E18a*x(20))/(KM18af*KM18ar+KM18ar*x(19)+KM18af*x(20)));
% Variation of NMol
GN*(kc17a*E17a*x(19)/(x(19)+KM17a)); % Variation of Mol
GN*(kc18b*E18b*x(18)/(x(18)+KM18b)); % Variation of IMol
GN*(kc17b*E17b*x(18)/(x(18)+KM17b))]; % Variation of NIMol
else t>= 1296000 %(patterns of enzymes from 15 - 40 days
after leaf initiation)
% Second peak of enzyme activity:
b2=1814400; % Defines the position of the center of the second
Gaussian peak for
enzyme activity.
% Relevant to the following enzyme activities: PR
c2=1420000; % Defines the width of the second Gaussian peak for
enzyme activity at
half maximum.
% Relevant to the following enzyme activities: PR
b4=2160000; % Defines the position of the center of the second
Gaussian peak for
enzyme activity.
% Relevant to the following enzyme activities: IsoDH, IsoR,
IsoI
c4=170000; % Defines the width of the second Gaussian peak for
enzyme activity at
half maximum.
% Relevant to the following enzyme activities: IsoDH, IsoR,
IsoI
E12=(0.044)*1*exp((-(t-b4).^2)/(2*(c4)^2)); % IsoDH
E13=(0.204)*0.0044*exp((-(t-b2).^2)/(2*(c2)^2)); % IsoR
E14=(0.204)*0.0044*exp((-(t-b2).^2)/(2*(c2)^2)); % IsoI
E16a=(0.204)*0.00014*exp((-(t-b2).^2)/(2*(c2)^2)); % PR
(product: (-)-menthone)
E16b=(0.204)*0.000014*exp((-(t-b2).^2)/(2*(c2)^2)); % PR
(product: (+)-isomenthone)
xdot=[GN*(-(kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2))));
% Variation of GAP
GN*(-(kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2))));
% Variation of Pyruvate (same expression as for GAP)
GN*((kc1b*E1*x(2)*x(1)/(Kia*KM1b+KM1a*x(2)+KM1b*x(1)+x(1)*x(2)))-((KM2r*kc2f*E2*x(3)-KM2f*kc2r*E2*x(4))/(KM2f*KM2r+KM2r*x(3)+KM2f*x(4))));
% Variation of DOXP
GN*(((KM2r*kc2f*E2*x(3)-KM2f*kc2r*E2*x(4))/(KM2f*KM2r+KM2r*x(3)+KM2f*x(4)))-(kc3*E3*x(4)/(x(4)+KM3)));
% Variation of ME4P
GN*((kc3*E3*x(4)/(x(4)+KM3))-(kc4*E4*x(5)/(x(5)+KM4))); %
Variation of CDP-ME
GN*((kc4*E4*x(5)/(x(5)+KM4))-(kc5*E5*x(6)/(x(6)+KM5))); %
Variation of CDP-ME2P
GN*((kc5*E5*x(6)/(x(6)+KM5))-(kc6*E6*x(7)/(x(7)+KM6))); %
Variation of MEcPP
GN*((kc6*E6*x(7)/(x(7)+KM6))-(kc7*E7a*x(8)/(x(8)+KM7))-
(kc7*E7b*x(8)/(x(8)+KM7))); % Variation of HMB-PP
GN*((kc7*E7a*x(8)/(x(8)+KM7))+(kc8f*E8*x(10)/(x(10)+KM8f*(1+(x(11)/Kic1))))-(kc8r*E8*x(9)/(x(9)+KM8r*(1+(x(11)/Kic1))))-((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b)));
%Variation of DMAPP
GN*((kc7*E7b*x(8)/(x(8)+KM7))+(kc8r*E8*x(9)/(x(9)+KM8r*(1+(x(11)/Kic1))))-(kc8f*E8*x(10)/(x(10)+KM8f*(1+(x(11)/Kic1))))-((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b)));
%Variation of IPP
GN*(((kc9a*E9*KM9b*x(9)+kc9b*E9*KM9a*x(10))/(KM9b*x(9)+KM9a*x(10)+KM9a*KM9b))-
(kc10*E10*x(11)/(x(11)+KM10))); %Variation of GPP
GN*((kc10*E10*x(11)/(x(11)+KM10))-(kc11*E11*x(12)/(x(12)+KM11))); %
Variation of LIM
GN*((kc11*E11*x(12)/(x(12)+KM11))-(kc12*E12*x(13)/(x(13)+KM12))); %
Variation of IPPol
GN*((kc12*E12*x(13)/(x(13)+KM12))-(kc13*E13*x(14)/(x(14)+KM13))); %
Variation of IPPone
GN*((kc13*E13*x(14)/(x(14)+KM13))-(kc14*E14*x(15)/(x(15)+KM14))); %
Variation of CIPUL
GN*((kc14*E14*x(15)/(x(15)+KM14))-(kc16a*E16a*x(16)/(x(16)+KM16a*(1+z*(x(17))/Kic2)))-(kc16b*E16b*x(16)/(x(16)+KM16b*(1+z*(x(17))/Kic2)))-(w*kc16a*E16a*x(16)/(KM16a+x(16)*(1+x(16)/Kis)))-(w*kc16b*E16b*x(16)/(KM16b+x(16)*(1+x(16)/Kis)))-(kc15*E15*x(16)/(x(16)+KM15)));
% Variation of PUL
GN*(kc15*E15*x(16)/(x(16)+KM15)); % Variation of MF
GN*((kc16b*E16b*x(16)/(x(16)+KM16b*(1+z*(x(17))/Kic2)))+(w*kc16b*E16b*x(16)/(KM16b+x(16)*(1+x(16)/Kis)))-(kc17b*E17b*x(18)/(x(18)+KM17b))-(kc18b*E18b*x(18)/(x(18)+KM18b)));
% Variation of IMone
GN*((kc16a*E16a*x(16)/(x(16)+KM16a*(1+z*(x(17))/Kic2)))+(w*kc16a*E16a*x(16)/(KM16a+x(16)*(1+x(16)/Kis)))-((KM18ar*kc18af*E18a*x(19)-KM18af*kc18ar*E18a*x(20))/(KM18af*KM18ar+KM18ar*x(19)+KM18af*x(20)))-(kc17a*E17a*x(19)/(x(19)+KM17a)));
% Variation of Mone
GN*((KM18ar*kc18af*E18a*x(19)-KM18af*kc18ar*E18a*x(20))/(KM18af*KM18ar+KM18ar*x(19)+KM18af*x(20)));
% Variation of NMol
GN*(kc17a*E17a*x(19)/(x(19)+KM17a)); % Variation of Mol
GN*(kc18b*E18b*x(18)/(x(18)+KM18b)); % Variation of IMol
GN*(kc17b*E17b*x(18)/(x(18)+KM17b))]; % Variation of NIMol
End
8. Statistical analysis of goodness of fit between simulated and
measured monoterpene profiles
The focus of this study was to evaluate the minimum set of
experimental data that, when incorporated into our model, led to
simulations reflecting experimentally determined monoterpene
profiles. It is important to note that we did not use any model
fitting approaches in the present work. All modeling assumptions
were inferred directly from experimental data. To assess the
validity of our modeling assumptions we performed a very basic
goodness of fit analysis. We did not attempt to accurately simulate
the entire time course of accumulation for each monoterpene. We
were primarily interested in how well we could simulate the
composition of monoterpenes at 40 d, which corresponds to the
developmental stage when commercially grown peppermint would be
harvested for oil extraction. For this analysis we selected the Chi
Square goodness of fit test. This test is usually employed when one
attempts to fit a statistical model to observed data, and one is
assessing how well the model actually reflects the data. In
general, the Chi Square test statistic is of the form
where
Oi = observed values (in the context of the present study this
refers to simulated data)
Ei = expected values (in the context of the present study this
refers to experimental data)
Although this test is commonly used to evaluate frequency
distributions (with very large sample sizes), it provided us with
an opportunity to apply statistical hypothesis testing to the
semi-quantitative evaluation of modeling results. The Chi Square
statistic can then be used to calculate a p-value with various
online tools (we use the tool at the following URL:
http://faculty.vassar.edu/lowry/tabs.html). This calculation
requires entry of degrees of freedom, which, in our study, is equal
to the number of analytes n (for the statistical analysis the
goodness of fit for 5 major p-menthane monoterpenes and a separate
“Other” category (all other monoterpenes) was determined; total of
6 analytes) minus “1”. In statistical hypothesis testing the null
hypothesis is generally rejected if the p-value is less than 0.05
or 0.01, corresponding to a 5% or 1% probability of obtaining a
certain result by chance. In the context of our experiments a
non-significant p-value (p > 0.05) means that there are no
statistically significant differences between experimentally
determined and simulated monoterpene composition at 40 d,
indicating a good fit between experimental and simulated data.