How Plants Function
(the basics of modeling)
Content
Equilibrium Partitioning
Equilibrium
Model for Roots
Translocation upwards
Model for Leaves
Model for Fruits
“Standard model”
Chemical Phases and Environment
Measurement in the laboratory
KOW Partition coefficient octanol – water
Laboratory
Environment
Gas, air
Atmosphere
S Water solubility (mg/L)
Water
Water
+ P Vapour pressure (Pa) = Henry’s Law constant KAW
Hydrophobic organic
phase
Lipids, Organic Matter,
Waxes, Cuticles, Skin
Glass
Stones, Sand
Meaning for chemical fate in the environment
Distribution between water, air, soil and lipids
Partition coefficient K
= Concentration ratio in phase equilibrium
Kij = Ci/Cj
K is partition coefficent (kg/m3 to kg/m3 = m3:m3)
C is equilibrium concentration (kg/m3)
i and j are indices of phases
KAW Partition coefficient between Air and Water
KOW Partition coefficient between Octanol and Water
KOC Partition coefficient between Organic Carbon and Water
1
Data Uncertainty
The values of physico-chemical properties (KAW, KOW
etc.) vary with method, lab, equipment and experience
of the staff. In particular extreme values (very high
KOW, low KAW) tend to be uncertain.
Sorption to Soil
Distribution coefficient Kd between dry soil and water
CM = Kd CW
[L/kg]
CM concentration sorbed to sediment matrix [mg/kg]
KSW = CS / CW = Kd S + PW
Try to use recent data; cross-validate the data with
property estimation methods (book chapter 11) or use
established estimation methods (ACD).
 is density [kg/L !]
PW is volumetric water content of the sediment [L/L]
Unpleasant unit (L / L) – soil is typically in kg
Sorption to Soil
Concentration in soil pore water
Distribution coefficient Kd between dry soil and water
CM = Kd CW
wet soil to water [m3:m3]
CW / CSoil = KWS = wet / (Kd x dry + PW)
[L/kg]
Unit: mg/L : mg/kg = kg/L
CM concentration sorbed to sediment matrix [mg/kg]
with Kd = OC * KOC
KSW = CS / CW = Kd S + PW
wet soil to water [L/L]
CW
ρ wet
1

 KWS 
CSoil OC  K OC  ρ dry  PW
K SW
 is dry density [kg/L !], PW is water content [L/L]
Unpleasant unit (L / L) – soil is typically in kg
CW in mg/L; CSoil in mg/kg bulk soil
OC in kg/kg;  in kg/L bulk soil; PW in L/L bulk soil
 divide by (wet) density
Estimation of the Kd-value
Regressions between log KOC and log KOW
Humic substances (organic carbon) are hydrophobic
10
8
KOC partition coefficient organic carbon to water
6
OC organic carbon content (g/g dry mass) also fOC, orgC
Regression between KOC and KOW
log Koc
Kd = KOC OC
4
2
0
-2 0
-5
log KOC = 0.81 log KOW + 0.1 (EU 1996, use this)
5
10
-4
log Kow
Older:
log KOC = 0.72 log KOW + 0.49 (Schwarzenbach & Westall 1981)
KOC = 0.411 KOW (Karickhoff 1981)
EU
Schwarzenbach
Karickhoff
Usually very small differences between the regressions
2
”Aging”
RCF or Partition constant Root to Water KRW
= equilibrium root to water
RCF
With time, pollutants in soil are ”sequestered” and get less
bioavailable.
Mathematically, this can be treated by two different ways:
log( RCF  0.82)  0.77 log K OW  1.52
● Irreversible loss (no more bioavailable; similar to
degradation)
● Increase in Kd (Koc) with time (stronger sorption, but partly
still bioavailable)
log Kow
W is 0.82
What would the audience suggest (discussion)?
Data by Briggs et al. (1982) for barley
log KSW and log KRW plotted versus log KOW
Definition
1.E+04
RCF = Root Concentration Factor =
1.E+03
log K
Concentration in Root (mg/kg)
――――――――――――――――――
Concentration in external solution (mg/L)
1.E+02
1.E+01
1.E+00
Briggs et al. (1982):
1.E-01
log (RCF - 0.82) = 0.77 log KOW - 1.52
or
RCF = 0.82 + 0.03 KOW0.77
-2
0
2
4
6
log Kow
KSW
Trapp (1995, 2002):
KRW = WR + LR a KOW b
KRW
Water content of roots higher than in soil
 polar compounds more in roots
with W R = 0.82 L/kg, a = 1.2 L/kg, L = 0.025 kg/kg and b = 0.77
Sorption to root lipids very similar to sorption to soil organic
carbon  same K to water
 identical results
Equilibrium Root to Soil
Leaf Concentration Factor
Root to water: KRW
0.81
K SW 
0.02  K OW  1.6  0.2
1.95
5
K root-soil
Soil to water:
Leaf to water
6
KRW = 0.85 + 0.03 KOW 0.77
KLW = WR + LR a KOW b
4
with W R = 0.8 L/kg, a = 1.22 L/kg, L = 0.02 kg/kg and b = 0.95
3
2
Leaf to air
1
0
Root to soil:
KRS = KRW / KSW
-1
1
3
5
7
log Kow
Root and soil organic carbon have similar
adsorption capacity  KRS close to unity
KLA = KLW / KAW
(Divide the partition coefficient leaves to water KLW by the partition
coefficient water to air KAW)
3
How Plants Function
Phase equilibrium
what is it ?
The equilibrium is the condition with the highest entropy.
Diffusion processes always go towards higher entropy, i.e.
to equilibrium. The smaller the scale, the closer the
concentrations are to equilibrium (local equilibrium).
book pg. 45, chapter 4.5
Water use of plants
How plants function
Plants do not ”use” the water that they
take up.
Roots take up water and solutes
Most of it is leaving the leaves again.
It is only required to keep the inner
leave wet when stomata are open to
take up CO2.
Stems transport water and solutes
Xylem = water pipe
Phloem = sugar pipe
The “water use efficiency” WUE is
growth (kg dry weight) per transpired
water (L).
Leaves transpire water
and take up gas
Fruits are sinks for phloem and
The transpiration coefficient TC (L/kg)
is L transpired water per kg dry
biomass produced = 1 / WUE.
xylem
Water use of plants
Herbal C3-plants
Cereals
Leguminosae
Potatoes
Woody plants
Tropical trees
Deciduous trees
Needle trees
C4-plants
maize, sorghum
Cacti (CAM plants)
TC (L/kg)
500 – 600
700 – 800
400 – 650
Transpiration of plants in Denmark
WUE (g/L)
1.5 – 2.0
1.2 – 1.4
2.5 – 1.5
600 – 900
200 – 350
200 – 300
1.6 – 1.1
5.0 – 2.9
5.0 – 3.3
220 – 350
50 – 150
4.5 – 2.9
20 – 6.6
Type
mm/year
mm/d
Broad-leaf trees
500-800
4-5
Needle trees
300-600
2.5-4.5
Corn fields
400-500
Pasture, meadows
300-400
3-6
General rule:
About 2/3rd of precipitation is transpired by plants.
For our typical crops, the variation in water use is small!
4
Basic data
Basic data used in the standard model
Example
Transpiration: 500 mm/year = 500 L/m2
Transpiration coefficient TC = 500 L /kg
 Biomass growth per m2 = 1 kg m-2 year-1 (dry weight)
Plant mass per m2
roots 1 kg (wet weight)
leaves 1 kg
fruits ½ kg
Transpiration
1 to 1.2 L d-1 m-2 (365 – 438 L/m2/year)
Compare to
- a good grain harvest is 60 dt/ha (6000 kg grain / 10 000 m2)
= 0.6 kg/m2 grain (wheat)
hereof 15% water  = 0.5
kg/m2
dry weight grain
Plus roots and straw (about half of the plant) gives 1 kg/m2 dw
Plant Uptake Models
Growth rate
0.1 d-1 (doubling in 1 week) for field crops
0.035 d-1 (doubling in 3 weeks) for meadows
Types of Chemicals
I Inorganic compounds
● Uptake pathways
Nutrients – active uptake, enzymatic regulation of uptake
● Model for roots
Heavy metals – eventually active uptake, facilitated
diffusion
● Translocation upwards
● Model for leaves
● Model for fruits
II Anthropogenic organic compounds
● “Standard” model
Typically passive uptake (advection & diffusion)
No involvement of enzymes
The following models are only for these chemicals !
Passive Uptake
Particle
deposition
Exchange with air
Roots
Xylem &
Phloem
transport
Direct
soil
contact
Diffusion
Translocation Soil – air
plant
in xylem
Advective uptake with water
5
Some facts about roots
► We can’t see them, but they are the most important part
of the plant, i.e. most plants can survive when you
remove stem and leaves – but rarely any without roots.
► Roots make up about 50% of the mass of the plant
(in extreme cases, such as desert plants, > 90%).
► The rhizosphere has the highest metabolic capacity of
the earth (roots, bacteria, fungi).
► Average maximum rooting depth is 2.1 m for cropland,
2.9 m for forest (humid zone), 4.6 m (globe).
The two big plant empires
Monocotyledone
Dicotyledone
(seedling with 1 leaf)
(seedling with 2
leaves)
grasses, corn,
onions, garlic, tulips,
lilies
All cereals
herbs, trees,
vegetables
All root vegetables
► Maximum observed rooting depth is 68 m (Kalahari).
Root structure
Monocotyledone
Root surface
Dicotyledone
To take up water and
solutes, plants make ”root
hairs” of a few m length.
These root hairs exist for
only a few hours and
increase the root surface
by factor 100 or more!
The calculated surface of 1 kg roots with 1 mm diameter is 4 m2
at a length of 1.27 km.
Roots start at one point, no
branching; thin, long
Fractal structure, branched
roots, fine roots and thick
roots.
The true surface – due to root hairs – is 1000 m2 at a length of 10 000 km of
one single rye plant (!).
Root surface
Consider this:
Root hairs only form in soil
or in water vapour – but
never in hydroponic
solution.
Model for Uptake into Roots
Thus, many uptake
experiments in water may
lead to artefacts.
6
Roots – Diffusive or advective uptake?
(Thick) Root model mass balance
High surface favours
diffusive exchange.
Change of mass in (thick) roots =
+uptake with water – transport to shoots
Root hairs have a high
surface.
 Root tips will always be
in chemical equilibrium to
soil (solution).
dmR/dt = CWQ – CXyQ
where
 The root solution flowing into the thicker parts of roots has the
same chemical concentration as the soil solution !
m is mass of chemical (mg)
C is concentration [mg/kg, mg/L]
Q is water flow [L/d]
All root vegetables are “thick roots” (of dicotyledones).
index R is roots, W is water and Xy is xylem
m is chemicals’ mass (mg)
M is root mass (kg)
C is concentration (mg/kg)
Dilution by exponential growth
Plant mass,
concentration
From mass to concentration
100
75
50
25
0
0
24
C=m/M
48
72
Time
M (kg)
dmR/dt = d(CR MR)/dt
The root grows – integration for C and M required
(oh no ...!)
Root model concentration
Change of concentration in roots =
+ uptake with water
– transport to shoots
– dilution by growth (rate k)
dCR/dt = CWQ/M – CXyQ/M – kCR
where
k is growth rate [d-1]
CXy is concentration in xylem = CR/KRW
CW is concentration in soil pore water
m/M (mg/kg)
Chemical mass: m = constant
Plant mass: M(t) = M(0) x e+kt
m/M = Concentration in plant: C(t) = C(0) x e-kt
Root model solution
Mass balance: change = flux in – flux out
dm
 CW  Q  C Xy  Q
dt
CW 
C Soil
Kd
Concentration: divide by plant mass M
dC CW Q C Xy Q


 kC R
M
M
dt
C Xy 
CR
K RW
Set to steady-state and solve for CR
0
CW  Q
CR  Q

 k  CR
M
K RW  M
CR 
Q
C
 soil
Q
 kM K SW
K RW
7
Comparison to experimental data
Root Model result for roots to soil
(Csoil = 1 mg/kg)
0.000
RCF
root model
For lipophilic compounds: growth dilution.
exp core
B1
80
53
equilibrium
KRW
PC
B1
2
Carrot
dynamic
model
38
8
PC
6
B5
4
log Kow
PC
2
PC
0
B2
8
0.0001
B1
TCE
0.001
0.010
0.001
01
BaP
0.01
0.100
PC
0.1
1.000
B1
1
PC
BCF carrot
(fresh weight)
C root (mg/kg ww)
10
exp peel
BCF > factor 100 below equilibrium
Translocation upwards in the xylem
A ”standard plant” transpires
500 L water for the production
of 1 kg dry weight biomass!
Translocation Upwards
= approx. 50 L per 1 kg fresh
weight
= approx. 1 L/day for 1 kg
plant mass
From outside into xylem
Definition TSCF
TSCF = ”Transpiration stream concentration factor”
TSCF = Conc in xylem sap / conc in soil solution [mg/L : mg/L]
If TSCF is high, good translocation upwards.
Two methods:
1) Regression to log KOW (Briggs et al., Dettenmaier et al.)
2) Calculation from root model
To enter the xylem, the chemical must pass
root cells and Casparian strip
8
Regression for TSCF by Briggs et al.
Other regressions for TSCF
Burken and Schnoor (1998)
 - (log K OW - 2.50) 2 
TSCF  0.756  exp 
 also bell-shaped curved
2.58


Dettenmaier et al. (2009)
TSCF 
11
11  2.6 log K OW
sigmoid curve
 - (log K OW - 1.78) 2 
TSCF  0.784  exp

2.44


Briggs et al. (1982) = optimum curve
Calculation of TSCF from root model
Calculated concentration in xylem
(from root model)
dC R
Q
Q

 KWS  C S 
 CR  k R  CR
dt
MR
M R  K RW
influx
CR
Q

Q
CW
 kM
K RW
C Xy 
Q
CR
K RW
C Xy 
outflux
K RW
 CW
Q
 kM
K RW
C Xy 
Q / K RW
CR

 CW
K RW Q / K RW  k  M
It can be seen immediately that when KRW is low
(KOW is low), CXy (TSCF) is high.
Comparison of TSCF methods
CXy / Cw
1.0
0.8
0.5
0.3
0.0
0
1
2
3
4
5
6
7
log Kow
Other data
1.2
TSCF
0.8
0.4
0
-1
1
3
5
log Kow
Briggs fit
B&S fit
TSCF calc
Dettenmaier
Briggs’ and B&S’ method give a bell-shaped curve.
Dettemnaier et al. and model give a sigmoid curve.
Compilation of literature values by
Dettenmeier & Doucette (2009)
9
All available data – what is truth?
Comparison of concentrations in xylem
CXy / Cw
1.0
0.8
0.5
0.3
0.0
-2
0
2
4
6
log Kow
Measured, pressure chamber
Dettenmeier et alii (2009)
Calculated
TSCF Briggs
Compilation of literature values by
Dettenmeier & Doucette (2009)
Sorption to wood
KWood = CWood / Cw
log KWood = – 0.27 + 0.632 log KOW
(oak)
log Kwood
log KWood = – 0.28 + 0.668 log KOW
(willow)
Lignin is a good sorbent
for lipophilic chemicals!
Chemicals in the Stem
Chemicals travel upwards with the water in the stem
● loss (volatilization, degradation, growth dilution)
● uptake from air
dC
 kt  I
dz
k: overall loss rate; I: gain from air
C ( z )  C (0)  e kz / u 
I
 (1  e  kz / u )
k
Stem is only considered in the Fruit Tree model
Uptake of contaminants into leaves
Leaves are highly exposed to air
Leaves
High water flux to leaves (xylem)
 Contamination possible from soil and air
10
Model for uptake into leaves
Mass balance: uptake from soil and air
+ - exchange with air
Outflux from roots is influx to shoots
dC R
Q
Q

 KWS  C S 
 CR  k R  CR
dt
MR
M R  K RW
is influx to leaves and fruits
dC L
Q

 CR
dt
M L  K RW
- dilution by growth
- metabolism
+ influx with xylem
Leaves – exchange with air
Remember: high for polar compounds (low log Kow)
Equilibrium between leaves and air
Leaves are plant material, like roots. But they do not
hang in soil, and not in water. Leaves hang in air.
The concentration ratio between air and water is
Stomata 
C Air
 K AW
CWater
The concentration ratio between leaves and air is then
Cuticle
C Leaves C Leaves CWater


 K LW / K AW  K LA
C Air
CWater C Air
Because KAW < 1 and KLW > 1  KLA >> 1
Example calculation: Equilibrium leaf-air for benzo(a)pyrene
leaf density  = 500 kg m-3, log KOW = 6.13, KAW = 1.35 x 10-5
Exchange with Air
Calculation with Fick’s 1st Law of Diffusion
Change of mass in leaves = + uptake from air - loss to air
KLW = W + L a KOW b
= 0.8 + 0.02 x 1.22 x 1 348 962 0.95 = 16251 L/L
(to water)
KLA = KLW/KAW = 16251 / 1.35 x 10-5
KLA/ = 1.2 x
109 (mg/m3
:
mg/m3)
= 1.2 x 109 (mg/m3 : mg/m3)
/ 500 kg/m3 = 2.4 x 106 (m3/kg)
(to air)
dmL
C
 ( A  g  C Air  A  g  L )
dt
K LA
A is leaf area [m2], g is conductivity (= exchange velocity 1 mm/s)
KLA is needed because the diffusion is between 2 different media
If CAir = 1 ng m-3 = 10-6 mg m-3
CLeaf = CAir x 2.4 x 106 m3 kg-1 = 2.4 mg kg-1 (chemical equilibrium)
11
Exchange with Air
Conductance versus Permeability
by
-gaseous deposition
through the cuticle
-gaseous exchange
through the stomata
- dry particulate deposition
- wet particulate deposition
A rough estimate for the
exchange velocity g is 1 mm/s
(default value)
Take care! Typically:
Conductance g is related to gas phase concentration
dmL
C
 ( A  g  C Air  A  g  L )
dt
K LA
Permeability P is related to concentration in water
C
P
dm L
 (A
 C Air  A  P  L )
K LW
K AW
dt
is equivalent, if g = P / KAW
Mass balance for the leafy vegetables
The model for leafy vegetables
The change of mass in leaves =
Adapted by the EU in the Technical
Guidance Documents for Risk
Assessment  ”TGD model”
+ translocation from roots + uptake from air - loss to air
from roots
growth &
degradation
to air
Uptake from soil (via xylem) and from
air (or loss to ...)
A  g 1000L m3
A g
dCL
Q
 CL  kL  CL

 CR  L
 CA  L
KLA  M L
ML
dt M L  KRW
+ Exponential growth
easy to solve: linear diff. eq. of the type
Extension: With particle deposition
fp is particulate fraction
1 – fp is gaseous fraction
AL  vdep
dC L
Q
A g

 CR  L
 (1  f P )  C A 
 fP  CA
2 M L
dt
M L  K RW
ML
AL  g L  1000 L m 3
 CL  k L  CL
K LA  M L
dC L
 I  kC L
dt
Dataset Leaves (1 m2)
Cair = total concentration

from air
Symbol
Value
Unit
Shoot mass
ML
1
kg
Leaf area
A
5
m2
kg/m3
Parameter
Shoot density

500 - 1000
Transpiration
Q
1
L/d
Lipid content
L
0.02
kg/kg
Water content
W
0.8
L/kg
Conductance
g
10-3
m s-1
Growth rate
kL
0.035
d-1
t
60
d
Time to harvest
12
Transfer to leaves with attached soil
Soil on plant
surfaces (Li et al.
1994)
[g soil/kg plant dw]
Lettuce
Wheat
Cabbage
260
4.8
1.1
Default value: 1% attached soil (wet weight)
Direct Soil Uptake
A ”standard” child eats
200 mg soil a day
”Pica child: 10 grams
(acute effects)
How much soil do you eat?
More than you think ...
(1% of 500 g/d is 5000 mg/d)
BCF(leafy vegetables to soil) = BCF model + 0.01
Fruits of all kind
provide the major part of our diet (bread, potatoes, juice)
Model for Uptake into Fruits
“Fruit model” very similar to leaf model but with
different parameterization
Model for uptake into fruits
Mass balance: uptake from soil and air
Fruit model (2009)
phloem +
xylem flux
from air
to air
dilution and
metabolism
dm F
Q
A  PF
A  PF
 F  CR  F
 C Air  1000  F
 C F  k met , F  m F
dt
K RW
K AW
K FW
+ - exchange with air
- dilution by growth
dC F
QF
A  PF
A  PF

 CR  F
 C Air  1000  F
 CF  k F  CF
dt
M F K RW
M F K AW
M F K FW
- metabolism
Solution
+ influx with xylem
and phloem
C F (t )  C F (0)  e kt 
I
(1  e kt )
k MF
+ 0.1% particle
attachment
13
Data for Fruits and Leaves
The “Standard Model” (2009)
Fruits
Leaves
Unit
kgww
Mass
M
1
1
Area
A
1
5
m2
Density

1000
500 1000
kgww m-3
Water stream
Q
0.2
1
L d-1
Lipid content
L
0.02
0.02
kg kgww-1
Water content
W
0.15
0.8
L kg-1
g
86.4
86.4
m d-1
vdep
86.4
86.4
m d-1
Growth rate
k
0.035
0.035
d-1
Time to harvest
t
60
60
d
Attached soil
R
0.001
0.01
kg/kg
Conductance
Deposition velocity
from air
dC R
Q
Q

 KWS  C S 
 CR  k R  CR
dt
MR
M R  K RW
AL  v dep
dC L
A g
Q

 CR  L
 (1  f P )  C A 
 fP CA
dt
M L  K RW
ML
2 M L

AL  g L  1000 L m 3
 CL  kL  CL
K LA  M L
dC F
QF
A  PF
A  PF

 CR  F
 C Air  1000  F
 CF  k F  CF
dt
M F K RW
M F K AW
M F K FW
Implementation in excel – free for all
Uptake from soil into leaves
1000
100
C Leaves
10
1
1
-1
-3
0.1
-5
0.01
-7
-7
0.001
-3
0.0001
-2
0
log Kaw
-9
1
2
4
6
partitioning
air-water
log Kow
Accumulation in leaves: polar, non-volatile compounds
(such as pesticides, detergents, pharmaceuticals)
Uptake from soil into fruits
Uptake into fruits from air
10
10
1
1
1
-1
0.1
C Fruit
0.01
0.001
-7
-3
0.0001
-2
0
2
0
C Fruits
-3
0.01
-5
0.001
2
4
6
-7
log Kaw
1
4
-2
0.1
6
log Kow
Accumulation in fruits: less than in leaves, but also
polar and non-volatile compounds
-9
2
0.0001
1
-1
-3
log Kaw
-5
-7
6
log Kow
-2
-9
“the usual candidates”: semivolatile lipophilic organic
compounds such as PCB, DDT, PAH, PCDD/F
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Bioaccumulation of lipophilic chemicals
Bioaccumulation of hydrophilic
compounds from soil in plants
A typical plant transpires 500 L water
for the production of 1 kg dry weight
biomass!
We all learned at university (did you ???):
”Lipophilic chemical accumulate via the food-chain”
= ~ 50 L per 1 kg fresh weight
high log KOW  high bioaccumulation
= ~ 1 L/day for 1 kg plant mass
The chemical comes with the water,
the water evaporates, the chemical
remains.
this is only one
out of two
mechanisms
This can lead to a bioaccumulation
plant to soil of up to factor 1000 (!)
BCF: Empirical regression by Travis & Arms
Easy to use
Gives good results
Something easier:
Implemented in RISK
The Regression of Travis & Arms
Problem: only uptake
from soil; no air
Regression with log KOW for C vegetation to C soil (dry wt.)
log BCF  0.578  log K OW  1.588
Travis and Arms, compared to Root model
(Csoil = 1 mg/kg)
C root (mg/kg ww)
10
1
0.1
BaP
TCE
0.01
Range of
soil
attachment
0.001
0.0001
0
2
4
6
8
log Kow
T&A
RCF
root model
Regression of T&A very similar to root model
Any questions?
15