4
Supplementary material to ‘Contribution of
Dissociation of Cd-EDTA Complexes to Cadmium
Uptake by Maize: a Modelling approach’
published in Plant and Soil
5
Jean-Marc Custos1, 2, Christian Moyne3, 4, Tiphaine Treillon1, 2, Thibault
6
Sterckeman1, 2*
1
2
3
7
8
1
9
40602, 54518 Vandœuvre-lès-Nancy Cedex, France
Université de Lorraine, Laboratoire Sols et Environnement, UMR 1120, TSA
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2
11
Vandœuvre-lès-Nancy Cedex, France
12
3
13
Appliquée, UMR 7563, BP 160, F-54504 Vandœuvre-lès-Nancy Cedex, France
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4
15
7563, BP 160, F-54504 Vandœuvre-lès-Nancy Cedex, France
INRA, Laboratoire Sols et Environnement, UMR 1120, TSA 40602, 54518
Université de Lorraine, Laboratoire d'Énergétique et de Mécanique Théorique et
CNRS, Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée, UMR
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17
*Corresponding author:
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Phone: + 33 (0)3 83 59 58 66
19
Fax: + 33 (0)3 83 59 57 91
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E-mail: Thibault.Sterckeman@univ-lorraine.fr
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Document with 21 pages, 4 tables and 9 figures
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1/20
24
Table of contents
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26
27
Supplementary material to ‘Contribution of Dissociation of Cd-EDTA Complexes to
Cadmium Uptake by Maize: a Modelling approach’ ............................................................... 1
28
I.
Analytical Derivation: 1-D plane steady state model in hydroponics .......................... 3
29
II.
Additional tables and figures ....................................................................................... 7
30
III.
References .............................................................................................................. 20
31
32
2/20
33
I.
34
Analytical Derivation: 1-D plane steady state model in
hydroponics
35
36
Let us consider that most of the assumptions on the parameters together with the
37
solutes concentrations at equilibrium were set previously in the main text.
38
Considering the solution-root system as a 1-D plane geometry in steady state,
39
transport equations are:
Dl
40
41
d2 [Cd] cond
 k a [Cd][L]  k dCdL [CdL]  0
2
dx
S-(1)
d2 [CdL] cond
Dl
 k a [Cd][L]  k dCdL [CdL]  0
2
dx
S-(2)
d 2 [L] cond
 k a [Cd][L]  k dCdL [CdL]  0
2
dx
S-(3)
Dl
42
43
If excess of ligand occurs over Cd i.e. [LT]/[Cdtot] >>1, the ligand concentration can
44
be assumed to be roughly uniform in the reaction zone and equal to its value [L]∞ in
45
the non rhizospheric zone at x = δ (i.e. bulk solution). By setting ka* = kacond[L]∞
46
where ka* is a constant, the problem can be written on the following form:
47
d2 [Cd] 
Dl
 k a [Cd] k dCdL [CdL]  0
2
dx
S-(4)
d2[CdL] 
 ka [Cd] kdCdL [CdL]  0
dx 2
S-(5)
Dl
48
49
The roots being assumed as a zero-sink for Cd, and CdL and L not being absorbed
50
the boundary conditions at the root surface at x = 0 are written as :
51
[Cd]  0, Dl
d[CdL]
0
dx
and Dl
d[L]
0
dx
S-(6)
3/20
52
The boundary conditions at the interface between the rhizospheric zone and that not
53
influenced by the root (i.e. bulk solution) at x = δ are:
54
[Cd]  [Cd]  , [CdL]  [CdL] 
and [L]  [L] 
55
with the quantities [ ]∞ being the solutes concentrations when x → ∞
56
Eqs. S-(4) and S-(5) lead to:
Dl
57
d2
dx 2
 k*
 [Cd] 
 [Cd] 

  Q
 avec Q   a *
[CdL]
[CdL]
  ka
 k dCdL 

k dCdL 
S-(7)
S-(8)
58
To solve the system above, the square matrix Q is diagonalized. Its eigenvalues λ
59
are:
k a*  
60
k
*
a
 k dCdL
k
CdL
d



    k a*  k dCdL   0
S-(9)
61
*
CdL
The eigenvectors associated to the eigenvalues 1  0 et 2  ka  kd are
62
respectively:
63
 k dCdL 
v 1   * 
 ka 
64
and
65
1
v 2   
  1
S-(11)
Q  P 1 N P
S-(12)
67
0
0


N  
*
CdL 
 0 ka  kd 
S-(13)
68
and
69
 k CdL
P 1   d *
 ka
70
As a result, Eq. S-(8) becomes :
66
S-(10)
1 
1
 *
 1  k a  k dCdL
1
 *
 ka
1
k
CdL
d



S-(14)
4/20
0
 [Cd]  [CdL]   0
  [Cd]  [CdL 
 *
  


CdL
*
CdL   *
CdL
k
[
Cd
]

k
[
CdL
0
k

k
k
[
Cd
]

k
[
CdL
d
a
d  a
d
 a
 

71
d2
Dl 2
dx
72
As the boundary conditions at x = δ, the solution is given by :
73
[ Cd]  [ Cd]   [ CdL ]  [ CdL ]   a 
74
(16)
75
[CdL]  [CdL]   b sinh 1  x 
k *a [Cd]  [Cd]   k CdL
d
   
76
by writing:
x

ω
77
S-(15)

 1

k a*  kdCdL
δ
Dl
S-
S-(17)
S-(18)
78
and where a and b are two constants, it follows that:
79
x
x 
 

[Cd]  [Cd]  a k CdL
  b sinh  ω1  
d 1 
 
   
S-(19)
80
x
x 
 

[CdL]  [CdL]  a k *a 1    b sinh  ω1  
 
   
S-(20)
81
The constants a and b are determined in respect to the boundary conditions at x = 0:
82
a kdCdL  b sinhω  [Cd]

*
 a ka  b ω coshω  0
83
The molar density flux of Cd at the root surface, φ, is expressed as:
S-(21)
84
ω cosh ω ka*  kdCdL 
d[Cd]
[Cd]
 Dl
d x x 0
 ka* sinhω  kdcondω coshω
S-(22)
85
k a*
cond
1  K CdL
[L]
k dCdL
[Cd]
[Cd]
φ  Dl

D
l

k a* tanh ω
 1  K cond [L] tanh ω
1  CdL
CdL
ω
kd
ω
S-(23)
φ  Dl
1
5/20
86
87
88
89
As the chemical equilibrium is respected at x = δ, the total Cd is obtained by:
cond
[Cdtot ]   [Cd]   [CdL]   [Cd]  1  K CdL
[L]  
S-(24)
Then, Eq. S-(22) becomes:
[CdT ]
1
φ  Dl
δ 1  K cond[L] tanh ω
CdL
ω
S-(25)
90
Finally, it is written as:
91

92
This clearly shows that the potential for the transport is the total Cd in hydroponics.
93
Assuming that the ligand is uniform through the solution is a good approximation
94
even at low amounts of added ligand ([LT]<[CdT) as the association terms becomes
95
small compared to that of dissociation.
[CdT ]
cond
 KCdL
[L]  tanh 

Dl
Dl

S-(26)
96
6/20
97
II.
Additional tables and figures
98
99
Table S1. Definition and values of the parameters used in the models.
Symbol
Definition
Values and references
CdL stability constant
18.2 M-1a
CaL stability constant
12.42 M-1a
MgL stability constant
10.57 M-1a
Apparent stability constant of Lads
16.40 M-1 b
k dCdL
Dissociation rate constant of CdL
1.8 × 10-4 s-1 ref 1
kdCaL
Dissociation rate constant of CaL
1.28 s-1 ref 2
kaCd, L
Association rate constant of Cd with L
k dCdL K CdL
k aCa, L
Association rate constant of Ca with L
kdCaL K CaL
cond
K CdL
Conditional stability constant of CdL
K CdL K CaL [Ca]
k acond
Conditional association rate constant of Cd with L
cond
K CdL
k dCdL
Diffusible Cd in soil
5.3 × 10-2 mol m-3
Diffusible Cd in soil
40.1 mol m-3
Diffusible Cd in soil
5.16 mol m-3
Solution volume fraction
0.2 m3 m-3
Cd buffer power
429.3 m3 m-3 c
Ca buffer power
9.8 m3 m-3 c
Mg buffer power
6.4 m3 m-3 c
Diffusion coefficient of Cd, CdL, CaL and free L in
free water
5×10-10 m2 s-1 ref 1
Diffusion coefficient of Cd, CdL, CaL and free L in
soil solution
 3 / 2 Dl
Medium volume
In hydroponics, 0.65×10-3 m3
logK CdL
logK CaL
logK MgL
log K

Lads
CdT 
CaT 
Mg T 
θ
bCd
bCa
bMg
Dl
D
*
V
In soil, 2.23×10-3 m3
Lg f
Final total root length
Measured for each plant
Lg 0
Initial total root length in soil
0.1 m
r0
Average root radius
Measured for each plant
r1
Thickness of the zone influenced by roots
V  Lgf
CdL, CaL and L absorbing power of root
8 × 10-8 m s-1
Cd maximal influx
4 × 10-10 mol m-2 s-1 ref 3
In hydroponics, 3 × 10-5 mol m-3
ref 3
α
I max
Km
Cd Michaelis-Menten constant
In soil, 1 × 10-4 mol m-3 ref 3
tf
Simulation time
In hydroponics, 1 dd
7/20
In soil, 26 de
β
Root growth rate
Lg f
 Lg 0  t f
100
101
L represents EDTA.
102
a
103
104
b
105
106
c
107
d
Simulations were run for one day as the exposure solutions were renewed each day.
108
e
Simulations were run from 2 days after the germination of maize seeds.
109
ref. 1: Degryse et al. (2006)
110
ref. 2: Carr and Swartzfager (1975)
111
ref. 3: Mullins and Sommers (1986)
log K are from MINTEQA2 Version 4.0 Database.
Estimated so that calculated dissolved Cd at equilibrium corresponds to measured dissolved Cd at 10
d.
Calculated as the exchangeable metals (in moles per unit of soil volume) ratio to dissolved metals (in
moles per unit of soil solution volume)
112
8/20
113
114
115
Table S2. Speciation of Cd and EDTA at equilibrium in the exposure solutionsa of
the experiment H1 calculated thanks to Eqs. 37, 38 and 39 with log KCd-EDTA = 18.2
and log KCa-EDTA = 12.42b.
116
Cd2+
Cd-EDTA
(µM)
(% added EDTA) (% added EDTA)
0
1
0
0
0.5
0.505
99
1
1
6.81 × 10-3 93.3
[EDTA]/[CdT]
2
5
10
Ca-EDTA
6.8
4.93 × 10
-3
49.8
50.2
1.24 × 10
-3
20
80
5.53 × 10
-4
10
90
117
118
a
Total Cd and total Ca concentrations were respectively 1 µM and 3 mM in all solutions
119
b
log K are from MINTEQA2 Version 4.0 Database.
120
121
122
123
Table S3. Cd-EDTA buffering at equilibrium in the exposure solutionsa of the
experiment H2 calculated thanks to Eqs. 37, 38 and 39 with log KCd-EDTA = 18.2 and
log KCa-EDTA = 12.42b.
124
CdT
Cd-EDTA
Cd2+
(µM)
(% CdT)
(nM)
0.05
99.52
1.21
40
0.1
99.75
1.23
81
0.25
99.90
1.24
201
0.5
99.95
1.24
402
0.75
99.97
1.24
603
1
99.98
1.24
804
Cd-EDTA/Cd2+
125
126
a
[EDTAT]/[CdT] = 5 and the total Ca, [CaT] = 3 mM in all solutions.
127
b
log K are from MINTEQA2 Version 4.0 Database.
128
9/20
129
Table S4. Main characteristics of the soil used in pot cultivation.
Particle size distribution (% dry soil weight)
Clay
10.7
Silt
25.8
Sand
63.5
pH (H2O)
6.8
-1
CaCO3, g kg
<1
-1
Organic C, g kg
14
P Olsen, g P2O5 kg-1
0.163
C/N ratio
10.5
-1
CEC, cmol+ kg
6.12
-1
Exchangeable cations, cmol+ kg
-1
Total Cd, mg kg
-1
Total Fe, g kg
Ca
5.46
Mg
0.597
K
0.7
Na
0.0341
Fe
<0.005
Mn
0.0364
Al
0.0237
0.163
11.3
130
131
132
10/20
1.E-02
{EDTA}T/{Cd}T = 0.2
Total Fe in soil solution (M)
{EDTA}T/{Cd}T = 0.5
{EDTA}T/{Cd}T = 1
{EDTA}T/{Cd}T = 5
1.E-03
1.E-04
1.E-05
0
133
5
10
15
20
25
Cultivation period (d)
134
Fig. S1: Iron concentrations measured in soil solution over the maize cultivation
135
period, for the different initial {EDTA}T/{Cd}T. Points are mean values and errors
136
bars represent standard deviations (n = 4).
137
138
11/20
{EDTA}T/{Cd}T = 0
Total Cd in soil solution (M)
1.E-03
{EDTA}T/{Cd}T = 0.2
{EDTA}T/{Cd}T = 0.5
1.E-04
{EDTA}T/{Cd}T = 1
{EDTA}T/{Cd}T = 5
1.E-05
1.E-06
1.E-07
1.E-08
0
139
5
10
15
20
25
Cultivation period (d)
140
Fig. S2: Cadmium concentrations measured in soil solution over the maize
141
cultivation period, for the different initial {EDTA}T/{Cd}T. Points are mean values
142
and errors bars represent standard deviations (n = 4).
143
12/20
Total dry matter (g plant-1)
8
7
6
5
4
3
2
1
0
0
144
0.2
0.5
1
5
{EDTA}T/{Cd}T
145
Fig. S3: Measured biomass production of maize as a function of the total EDTA to
146
total diffusible Cd ratio, {EDTA}T/{Cd}T. Error bars are standard deviation (n = 4).
147
13/20
Fe in roots (mmol kg-1 DW)
160
140
a
120
100
80
60
40
20
0
0
0.2
0.5
1
5
1
5
{EDTA}T/{Cd}T
148
Fe in shoots (mmol kg-1 DW)
2.5
b
2.0
1.5
1.0
0.5
0.0
0
0.2
0.5
{EDTA}T/{Cd}T
149
150
Fig. S4: Roots (a) and shoots (b) Fe concentration measured in maize grown for 28 d
151
in a soil containing 0.037 mmol of diffusible Cd kg-1(4.2 mg kg-1) with increasing
152
EDTA to total diffusible Cd ratios ({EDTA}T/{Cd}T). Error bars correspond standard
153
deviation (n = 4).
14/20
Cd in roots (µmol kg-1 DW)
3 500
3 000
a
2 500
2 000
1 500
1 000
500
0
0
0.2
0.5
1
5
1
5
{EDTA}T/{Cd}T
154
Cd in shoots (µmol kg-1 DW)
70
b
60
50
40
30
20
10
0
0
0.2
0.5
{EDTA}T/{Cd}T
155
156
Fig. S5: Roots (a) and shoots (b) Cd concentrations measured in maize grown for 28
157
d in a soil containing 0.037 mmol of diffusible Cd kg-1 (4.2 mg kg-1) with increasing
158
EDTA to total diffusible Cd ratios ({EDTA}T/{Cd}T). Error bars correspond to
159
standard deviation (n = 4).
15/20
Cd uptake (µmol plant-1)
1.00
free Cd
0.80
CdEDTA
free Cd + CdEDTA
0.60
0.40
0.20
0.00
0
160
4
8
12
16
20
24
28
Time (d)
161
Fig. S6: Simulated evolution of Cd uptake components over the cultivation time.
162
Simulations were run from the germination of seeds i.e. 2 days after sowing. Final
163
total root length, average root radius and the thickness of the zone influenced by root
164
are respectively Lgf = 2822 cm, r0 = 0.014 cm and r1 = 0.5 cm.
165
16/20
Cd root influx (µmol (cm2 fresh root)-1 s-1)
1E-08
Measured
Simulated: kd = 1,8e-04 s-1; zero-sink
Simulated: kd = 1,4e-03 s-1; zero-sink
Simulated: kd = 1,8e-02 s-1; zero-sink
9E-09
8E-09
7E-09
6E-09
5E-09
4E-09
3E-09
2E-09
1E-09
0E+00
0.0
0.2
0.4
0.8
0.6
Total Cd (µM)
1.0
1.2
166
167
Fig. S7: Cadmium uptake of 23 day-old maize plants whose roots were exposed for 4
168
days to solutions containing increasing concentrations of Cd-EDTA complexes. Cd-
169
EDTA complexes accounted for more than 97% of total Cd in solution. Red
170
diamonds correspond to the measured uptake while the other symbols correspond to
171
simulated uptakes with a zero-sink for free Cd absorption and different values for the
172
dissociation rate constant (kd). The zero-sink influx is averaged on the overall root
173
surface, by using average values measured for the root length and radius over
174
experiment H2, which were Lgf = 890 cm and r0 = 0.118 mm, respectively.
175
176
17/20
177
4.5E-08
Cd uptake (µmol cm-2 s-1)
4.0E-08
3.5E-08
3.0E-08
Free Cd symplastic uptake
2.5E-08
Cd-EDTA uptake
2.0E-08
1.5E-08
1.0E-08
5.0E-09
0.0E+00
0.0
0.2
0.4
0.6
0.8
1.0
[Cd]T (µM)
178
179
Fig. S8: Theoretical Cd uptake by maize at low total Cd concentrations in the
180
presence and in absence of Cd-EDTA complexes. The solid and dotted black lines
181
represent the uptake of Cd2+ and that of Cd-EDTA complexes (as fitted in Figure 2),
182
respectively. Imax = 4.10-10 mol m-2 s-1 and Km = 3. 10-5 mol m-3 from Mullins et
183
Sommers (1986).
184
18/20
3.0E-04
Concentration (M)
2.5E-04
2.0E-04
Total soluble Cd
1.5E-04
[CdEDTA]max={Cd}T/theta
1.0E-04
5.0E-05
0.0E+00
0
185
100
200
300
400
500
{EDTA}T/{Cd}T
186
Fig. S9: Simulated evolution of total dissolved Cd as a function of total EDTA to
187
diffusible Cd ratio, {EDTA}T/{Cd}T. Solid line represents the maximal total
188
dissolved Cd {Cd}T/θ, that can be reached if all exchangeable Cd was desorbed.
189
19/20
190
191
192
193
194
195
196
197
198
199
200
III.
References
Carr J D and Swartzfager D G 1975 Kinetics of the ligand exchange and dissociation
reactions of calcium-aminocarboxylate complexes. Journal of the American
Chemical Society 97, 315-321.
Degryse F, Smolders E and Merckx R 2006 Labile Cd complexes increase Cd availability to
plants. Environ. Sci. Technol. 40, 830-836.
Mullins G L and Sommers L E 1986 Cadmium and zinc influx characteristics by intact corn
(Zea mays L.) seedlings. Plant Soil 96, 153-164.
20/20