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A Mathematical Model of Chenopodium album L.
Dynamics under Copper-Induced Stress
Laura R. Gonz´alez-Ram´ıreza,∗
, Deniz Ala¸camb,c , Aysegul Akpinard
aInstituto Polit´ecnico Nacional, Escuela Superior de F´ısica y Matem´aticas, Unidad
Profesional Adolfo L´opez Mateos Edificio 9, 07738, Cd. de M´exico, M´exico
bBursa Uluda˘g University, Col lege of Arts and Sciences, Department of Mathematics,
16059, Bursa, Turkey
cTri-institutional Center for Translational Research in Neuroimaging and Data Science
(TReNDS), Georgia State University, Georgia Institute of Technology, Emory University,
30303, Atlanta, GA, USA
dBilecik Seyh Edebali University, Biotechnology Application and Research Center, 11230,
Bilecik, Turkey
Abstract
Heavy metal contamination of the soil is a global problem that produces dif-
ferent harmful effects from an environmental and public health perspective.
Although there have been numerous efforts to solve this problem, there is no
precise methodology to decontaminate heavy-metal polluted soils. One of the
strategies to develop such methods relies on mathematical modelling. Pursuing
this goal, we propose a novel mathematical compartmental model consisting of
a linear system of differential equations to address the suitability of the model
plant (Chenopodium album L.) to be used in the remediation of contaminated
areas, such as sewage sludge lagoons. Our results show a tendency to maintain
high concentrations of copper (Cu) in the roots with the possibility of contin-
uing with good plants’ dynamics. Moreover, the model theoretically proposes
contaminant concentration in the plants’ shoots and roots and predicts a more
prolonged tendency to accumulate copper concentrations in the shoots and dis-
rupt the shoots’ dynamics. These results provide complementary support for
the suitability of this model plant to be used in contaminated areas. In ad-
dition, we present asymptotic tendencies of the plants’ biomass content and
∗Corresponding author
Email address: lrgonzalezr@ipn.mx (Laura R. Gonz´alez-Ram´ırez)
Preprint submitted to Ecological Modelling March 22, 2022
nitrogen-assimilatory (Nitrate reductase; NR) enzyme activity. In this way, we
project the relationship between contaminant accumulation and plants’ mea-
surements. These projections are important as they can potentially be used for
optimization purposes and strategic harvesting planning. Finally, we present a
parameter sensitivity analysis to complement the model examination.
Keywords: mathematical modelling, Chenopodium album L.; copper-induced
stress, sewage sludge lagoons, metal contamination, remediation
1. Introduction
Heavy metal contamination produces harmful effects on plant development,
and it is also an environmental threat of great magnitude for all living be-
ings. Growth inhibition is the most common change observed in plants ex-
posed to high heavy metal(s) concentrations (Prasad, 2004). In Gajewska and5
Sklodowska (2010), the authors stated that the primary plant part interacting
with heavy metals is the root system. Therefore, the decrease in growth is ob-
served more clearly in the roots than in the shoots. Heavy metals are taken
into the cell through the roots’ nutrient uptake pathways. Consequently, heavy
metals may compete with essential elements’ plant absorption. Therefore, the10
inability of the plant to absorb crucial nutrients may cause harmful effects on
cellular structures, primary metabolism, and transport processes (Boojar and
Goodarzi, 2007; Nedjimi and Daoud, 2009; Sharma and Dietz, 2009).
Nitrogen is one of the main structural components of the plant and constitutes
1.5-2% of the dry weight of the plant (Frink et al., 1999). Therefore, it is cru-15
cial to infer nitrogen concentration in plants through different measurements.
Plants usually take up nitrogen as nitrate (NO−
3). In order to fulfill the essential
metabolic functions, nitrate taken by the roots must first be reduced to nitrite
and then to ammonia. Nitrate Reductase (NR) and Nitrite Reductase (N iR)
enzymes actively reduce nitrate to ammonia. In particular, N R is one of the20
most important enzymes to infer the predominant form of nitrogen in plants
(Marschner, 1995; Solanki and Dhankhar, 2011). N R is highly sensitive to the
2
presence of nitrates. For this reason, it is suggested that N R activity reflects
the nitrate content of the habitat where the plant is located (Lee and Stewart,
1978). Indeed, N R activity is accepted in ecological studies as an indicator of25
nitrate presence (Gebauer et al., 1988; Olsson and Falkengren-Grerup, 2003).
NiR enzyme comes into play when NR enzyme converts nitrogen in nitrate
form to nitrite. In particular, NiR is responsible for the reduction of nitrite to
ammonia. With the assimilation of ammonia that occurs as a result of nitrogen
fixation, the nitrogen source needed by the plants is converted to glutamine and30
glutamate by specific reactions, and amino acids and protein compounds are
formed from these molecules (Zheng-Xun et al., 2007; Moschou et al., 2012).
Scientific evidence shows adverse effects of increased heavy metal concentration
in the growing environment, plant development, and water (Malec et al., 2008,
2009). Due to this fact, various physiological, biochemical, and molecular mark-35
ers have been established to determine the effects of heavy metal contamination
(Malec et al., 2008, 2009; Maleva et al., 2009; Wan et al., 2011; Kumar et al.,
2012). In this work, we are particularly interested in alterations from these
markers under Cu contamination. Some of the adverse effects on plants due
to Cu pollution have already been established. For example, the change in the40
plant’s NR activity is a consequence of the decrease in the plant’s nitrate uptake
due to copper pollution (Llorens et al., 2000; Xiong et al., 2006). In addition,
a reduction in NR enzyme activity may also occur due to the tendency of re-
active Cu ions to bind to the sulfhydryl compounds (SH) contained in the N R
enzyme. This state causes structural deterioration of the N R enzyme, resulting45
in irreversible consequences. Similarly, the NR enzyme may deteriorate struc-
turally as a result of low molybdenum (Mo) intake from micronutrients due to
high Cu concentrations. Specifically, Mo is a cofactor in the structure of NR,
and its insufficient uptake reduces NR enzyme activity.
Phytoremediation, which is also called a “green solution”, has a strategic po-50
tential to address the heavy metal pollution problem in the environment, and it
has been implemented worldwide (Swaileh et al., 2004; Zeidler, 2005; Gonz´alez
and Gonz´alez-Ch´avez, 2006; Ali et al., 2013). Phytoremediation aims to re-
3
duce contaminant’s concentrations or toxic effects in the environment by us-
ing heavy metal-tolerant plants and related soil bacteria (Greipsoon, 2011; Ra-55
jkumar et al., 2013). Scientific studies have shown that some plants, such as
Chenopodium album L., can survive in sewage sludge lagoons (Akpinar, 2021)
or heavy metal contaminated areas (Mohan et al., 2019; Bhargava et al., 2007;
Tozser et al., 2019; Alipour et al., 2015; Zulfiqar et al., 2012; Gupta and Sinha,
2007). Therefore, Chenopodium album L. is a candidate for monitoring heavy60
metal pollution in degraded areas (biomonitoring) and phytoremediation of
heavy metals in contaminated sites.
Hand-in-hand studies of theoretical and experimental researchers play a signif-
icant role in unveiling the underlying working principles of biological mecha-
nisms. Mathematical modelling is the primary tool used in theoretical studies.65
This approach provides a multidisciplinary framework for complementing, sup-
porting, and producing new experimental settings of a system’s dynamics. A
mathematical model can potentially project long-term behavior under different
physical constraints. On the one hand, a specialized model can include specific
factors and relationships of the system at the cost of increasing complexity of70
the model and hence limiting incorporation of experimental data. On the other
hand, a less complex mathematical model, so-called a “toy” model, can attenu-
ate this complexity at the cost of missing specific features of the system. A toy
model is primarily used for understanding the big picture more than the details.
Therefore, a possible approach in mathematical modelling is to incorporate a75
minimum set of system features without losing the model tractability of the
main variables. Additionally, one of the options when establishing a mathe-
matical modelling strategy is to develop a compartmental model. The basis of
compartmental models is to divide the system into multiple compartments to
be analyzed independently, which is supported by the so-called modular plant80
architecture approach. However, the meristematic nature of plants restricts the
assertiveness of this approach (Cheeseman et al., 1996). This is because there
is no exact distinction between a root and a shoot compartment in an actual
plant. Nevertheless, this simplifying assumption can build up basic models to
4
extract a set of the main dynamics that can be further refined later. In partic-85
ular, a linear mathematical model provides a simple scenario to investigate the
otherwise complex dynamics of a system.
Various mathematical models have been proposed to investigate and optimize
phytoremediation techniques. We found a model for the phytoremediation of
petroleum-contaminated soil (Thoma et al., 2003), an optimization model to90
comply with EPA criteria (Thomas and Vandemuelebroeke, 2005), a model for
the accumulation of heavy metals in benthic algae (Seip, 1979), among many
others (Cˆardei et al., 2021; D’Acunto et al., 2019; Torres-Bejarano et al., 2019;
Tantemsapya et al., 2011; Tawfiq and Wirojanagud, 2016). We also found a
heuristic mathematical model to assess a contamination level or plan a har-95
vesting strategy in these models. However, as far as the authors know, there is
currently no mathematical model relating both the N R enzyme activity and the
biomass content to the accumulation of contaminants in a plant. We propose
that incorporating these feature dynamics into a mathematical model is crucial.
Therefore, this approach relates contaminant accumulation to measurable exper-100
imental quantities intrinsic to the plant’s dynamics. The model here developed
aims to theoretically support the plausibility of the model plant to be used in a
copper-contaminated environment by relating the model to experimental data.
This work shows different scenarios of the main variables under copper-induced
stress. Furthermore, our results provide model plants’ consistent features with105
qualities observed in indicator species. In addition, our mathematical model as-
sesses and predicts the NR enzyme activity, biomass content, and contaminant
level in each of the plant’s compartments as time evolves.
The work in this manuscript is developed as follows. In Section 2, we estab-
lish the experimental analysis of Chenopodium album L. under copper-induced110
stress. Section 3, sets the biological and mathematical fundaments that sup-
port our mathematical model. Section 4 establishes the mathematical model
and analyzes different scenarios of the model plant under copper-induced stress.
Our model parameters are calibrated and validated by the experimental data.
Using these parameters we provide longer time projections of the variables. In115
5
addition to our forecasts, we provide asymptotic dynamics of the model’s main
variables, including information about the system’s more extended time dy-
namics. In Section 4, we also provide parameter sensitivity analysis to address
the model components and complement our mathematical analysis. Finally, in
Section 5, we discuss some of the model’s implications and limitations, and in120
Section 6, we establish our conclusions and future work to be developed.
2. Materials and Methods
Chenopodium album L., which is seen to grow naturally in sewage sludge lagoons
(Akpinar, 2021), was chosen as the model plant in this study. Uniform seedlings
belonging to the C. album were collected from their natural environments in125
Turkey’s Bursa Province. They were exposed to different concentrations of Cu
(0 µM (control), 50 µM, and 500 µM). Plants were harvested on the 1st, 3r d, and
7th days after the Cu treatments. Then, they were washed thoroughly with de-
ionized water, and the roots and shoots were separated (leaves+stems). The NR
enzyme activity was immediately analyzed in fresh plant material. NR in the130
fresh roots and shoots was determined according to the in vivo test described by
Hageman and Hucklesby (1971); Jaworski (1971), and modified by Gebauer et al.
(1984). This spectrophotometric method is based on measuring the absorbance
of nitrite (NO−
2) formed by the nitrate reduction in the incubation medium.
NR (µMNO−
2g−1DW h−1) was calculated using the absorption value on a135
dry weight (DW ) basis. Dry biomass was measured after oven-drying plant
materials at 80oCuntil constant weight. The biomass of roots and shoots was
also measured on a DW basis. The same method indicated in Akpinar (2021)
was used to determine Cu accumulation in plant parts.
3. Theory140
In this section, we provide the theory that supports the development of our
mathematical model. To begin, we design a compartmental setting. The com-
partments are determined by the environment E, the roots R, and the shoots
6
S, as is described in Figure 1. We aim to incorporate the dynamics of a metal
contaminant into each of these compartments in a simplified setting. Therefore,145
we assume that the independent compartments are only affected by the metal
contaminant. We establish a linear system of 9 first-order differential equations
with 13 free parameters that describe different plant dynamics. Although we
greatly simplify the biological process by considering a simple linear model, we
remain with the challenge of determining a plausible range for the model pa-150
rameters to study the effect of copper pollution on the model plant. Therefore,
we determine the parameters of our mathematical model by the experimental
data described in section 2. This provides a setting in which we theoretically
explore the suitability of the model plant, Chenopodium album L., to be used
in contaminated areas and assess the plant’s contaminant concentration. Our155
main variables are determined by the biomass and the nitrogen-assimilatory en-
zyme (NR) activities. The latter measurement indirectly provides information
about the plant’s nitrogen (N). Indeed, as previously mentioned, N R has been
established as one of the most important enzymes to determine the predominant
form of nitrogen in plants. In addition, the dynamics of the nitrogen content and160
the NR enzyme activity had been successfully modelled as oscillatory in higher
plants (Yang and Midmore, 2005), alluding to circadian oscillations. Therefore,
we propose an initial model of N R enzyme activity described by a second-order
constantly forced damped harmonic oscillator (Strogatz, 2015). To simplify the
model simulations, we transform the second-order differential equation into two165
first-order differential equations. That is, we consider:
dNR
dt =T
dT
dt =d−bT −aNR,
(1)
where T=dNR
dt is the rate of change of NR enzyme activity, and a,b,care
constants such that a > 0, b≥0, and d > 0. We propose that the rate of
change in nitrogen content (N C) is proportional to the rate of change in nitrate
reductase enzyme activity:170
7
dNC
dt =k1
dNR
dt ,(2)
for some constant k1>0. Furthermore, we propose that the biomass (B)
content is determined by the dynamics of the nitrogen content, which is directly
related to the amount of the NR enzyme in each compartment. Specifically, we
propose that the biomass is determined by the difference between the amount
of nitrogen and the decaying growth (Verkroost and Wassen, 2005), that is:175
dB
dt =k2NC −k3B, (3)
for some constants k2>0 and k3>0. Since we are primarily concerned with
the NR enzyme activity, we combine equations (2) and (3), and we propose the
equation:
dB
dt =K1NR −K2B, (4)
for some constants K1and K2.
The contaminant dynamics of the model are motivated by the work of Thomas180
and Vandemuelebroeke (2005). A mathematical model is established in the
abovementioned work to optimize harvesting strategies for metal phytoremedia-
tion. However, compared to Thomas and Vandemuelebroeke (2005), we directly
relate the contaminant accumulation to the plant’s intrinsic dynamics. Finally,
we assume that the metal accumulation directly affects the nitrogen content as185
a negative linear forcing term. That is, we propose the following equation to
describe the amount of the NR enzyme content under copper-induced stress:
dNR
dt =T
dT
dt =d−bT −aNR −eM,
(5)
where Mis the amount of metal absorbed by the plant and eis the uptake rate
of the contaminant.
We limit our analysis to the previously described features in the present model.190
8
(Shoots)
NS, BS
(Roots)
NR, BR
(Contaminant
in the envi-
ronmnent)
E
r1
r3
r2
Figure 1: Simplified compartmental flowchart of contaminant dynamics. NSand NRare the
amount of NR enzyme activities in the shoots and the roots, respectively. BSand BRare the
biomass content in the shoots and the roots, respectively. Under this scenario, the sum of the
amount of contaminant in the compartments remains fixed. The contaminant uptake rate in
the roots (r1), the shoots (r2), and return rate to the environment (r3) are almost equivalent
according to the experimental data.
A future version of the model should include different elements, such as carbon
dynamics, as carbon is one of the plants’ essential structural components. An
additional model description including specific information about water content,
photosynthetic activity, temperature, humidity, pH, among others, should also
be addressed in the future.195
The main assumptions and variables of the model are summarized as follows:
The model establishes two independent compartments of the plant de-
termined by the plant roots and shoots that are independently affected
9
by the contaminant. There is an additional compartment determined by
the plant growing conditions which are simplified and described as the200
“environment”.
We exclude the intrinsic dynamics of the environment.
We assume that a single contaminant dose is initially added to the envi-
ronment.
The main variables of the model are the NR enzyme activity, biomass con-205
tent, and contaminant concentration in both the roots and the shoots. The
experimental data constrains the NR enzyme activity and the biomass
content variable. In addition, the contaminant concentration variable is
constrained by the initial amount of contaminant added to the environ-
ment at the beginning of the experiment.210
The NR content is modelled as a damped harmonic oscillator. The forc-
ing of this oscillator accounts for all external factors not considered in
the modelling process. This is a central simplifying assumption aimed to
minimize the number of variables in the system.
The biomass content is related to the nitrogen content.215
The amount of metal in the environment affects the plant’s roots under
the law of mass action.
The amount of contaminant in the system remains fixed at all times.
An amount of metal is taken up directly by the roots, and another per-
centage goes immediately to the shoots.220
A portion of the contaminant not absorbed by the shoots is returned to
the environment.
All of the model parameters are assumed positive. These parameters
are not measured from experimental data. Instead, they are determined
10
by solving an optimization problem in which the NR enzyme activity,225
biomass content, and contaminant content variables fit the data.
4. Results
4.1. Mathematical Model
Inspired by the experimental studies, we establish a model to capture the rela-
tionship between contaminant uptake, biomass content, and NR enzyme activ-230
ity. The model is developed considering the simplifying assumptions previously
established. To simplify notation, we denote the NR enzyme activity variable
simply as Nj(t). We also denote Bj(t) as the biomass content, Tj(t) as the rate
of change of the Njvariable over time (i.e., Tj(t) = dNj(t)
dt ), and Mj(t) as the
amount of contaminant. All of the previous variables are evaluated at time t,235
and the subscripts j={R, S, E}denote the different compartments (R= Roots,
S= Shoots, and E= Environment). The time is measured in days. All of the
parameters are assumed positive, and their biological significance is described
in Table 1. The proposed model is:
dNR
dt =TR
dTR
dt =dR−bRTR−aRNR−eRMR
dBR
dt =αRNR−βRBR
dNS
dt =TS
dTS
dt =dS−bSTS−aSNS−eSMS
dBS
dt =αSNS−βSBS
dME
dt =−(r1+r2)ME+r3MR
dMR
dt =r1ME−r3MR
dMS
dt =r2ME.
(6)
Additionally, the model’s initial conditions are determined by the experimental
initial conditions in each of the different scenarios herein presented. Considering
11
the last three equations, we observe that:
dME
dt (t) + dMR
dt (t) + dMS
dt (t) = 0,(7)
implying that the amount of contaminant in the three compartments remains240
fixed at all times. That is, ME(t) + MR(t) + MS(t) = N, where Nis the total
amount of contaminant in the system.
Parameter Description Units
aRNR harmonic proportionality constant in the roots 1/days2
dRconstant environmental forcing in the roots DW/days2
bRNR damping term in the roots 1/days
αRNR influence on the biomass rate in the roots 1/days
βRbiomass influence rate in the roots 1/days
eRcontaminant influence rate in the roots DW/µM days2
aSNR harmonic proportionality constant in the shoots 1/days2
dSconstant environmental forcing in the shoots DW/days2
bSNR damping term in the shoots 1/days
αSNR influence on the biomass rate in the shoots 1/days
βSbiomass influence rate in the shoots 1/days
eScontaminant influence rate in the shoots DW/µM days2
r1contaminant uptake rate in the roots 1/days
r2contaminant uptake rate in the shoots 1/days
r3contaminant return rate to the environment 1/days
Table 1: Parameter description. The dry weight (DW ) units are measured in mg/kg.
We consider the proposed model under different parameter scenarios motivated
by the experimental data. Under convenient parameter choices, a mathemat-
ical model can make either quantitative or qualitative predictions. We focus245
on quantitative predictions; however, we will show that the metal accumulation
projections of our model might also be used to determine qualitative dynamics.
The parameters of each of the scenarios studied here were obtained by solving
12
Parameter A B C D E F
aR0.4153e42.5127 10.2825 4.1557 0.5793 1.7808
dR0.4514e43.2323 14.7604 5.1497 0.0150 1.0686
bR0.8017e43.6596 14.4398 0.1058 7.7625 3.6387
αR0.1110e40.3062 7.9737 1.8582 0.0003 1.8735
βR0.2923e41.0420 32.3239 6.4821 0.1261 4.2359
eR0.0008e40.0058 0.0340 0.0051 0.0001 0.0021
aS0.2743e40.0812 0.6782 0.5174 1.1344 0.3853
dS0.5556e40.9005 1.4521 1.0683 0.1105 0.3705
bS1.0410e40.0001 2.4012 0.5129 6.0540 0.0010
αS0.0198e40.0236 0.4355 0.0716 0.0362 1.9000
βS0.0533e40.0469 1.2148 0.2008 0.0420 3.6882
eS0.1846e40.0150 0.0080 0.0002 0.0001 0.0008
r10.3993e40.0556 0.0648 0.0524 0.0198 0.0204
r20.0000e40.0349 0.0342 0.0314 0.0062 0.0062
r32.5915e40.0001 0.0001 0.0001 0.1755 0.1650
Table 2: Parameters fixed for different model simulations motivated by experimental data.
Subject Ais a control experiment. Subjects B,C, and Dreceived 50 µM of C u. Subjects E
and Freceived 500 µM of Cu.
an optimization problem defined by a higher-order interpolation of the experi-
mental data. The optimization problem was solved using the MATLAB fmincon250
function (MATLAB, 2017). In all of the scenarios shown here (Subjects A-Fin
Table 2), we calibrate the model parameters by considering the first five days
(Subjects A,B,C, and D) or six days (Subjects Eand F) of the experimental
data. The remaining days of the experiment were used to validate the model’s
results. Since we aim to observe the effects of the contaminant in the system’s255
dynamics, we use the model to project more extended time dynamics after the
experiment. It is important to remark that since our model parameters are cal-
ibrated by data interpolation, future experimental settings that provide more
13
satisfactory time resolution can give a more accurate description of the system’s
dynamics.260
In Figures 2-7, we consider the dynamics of different experiments under Cu
contaminant levels. Each figure consists of a series of subfigures to describe the
system’s dynamics in the following way. In subfigures (a)(b)(e), we observe
the model calibration and validation periods. In subfigures (c)(d)(f), we fol-
low the projected long-term 14-days dynamics of the model variables. In all the265
subfigures, the experimental data are shown in red, and the model simulations
are shown in blue.
In Figure 2, we consider the dynamics of the control experiment (Subject A).
In this case, no contaminant is added to the plant growing condition. We ob-
serve a good resemblance between the experimental data and the model during270
the 7-day calibration and validation period. In Figures 2(a)(b), we observe a
slightly oscillatory behavior of the data, especially noticeable in the N R enzyme
activity in the shoots. Further refinement in experimental settings may provide
a more accurate description of oscillatory dynamics. In Figures 2(c)(d), we ob-
serve the projected 14-days dynamics of activities. In this control scenario, we275
observe a tendency of the shoots’ biomass and NR enzyme activity to remain
around “healthy” states. In the case of the roots variables, slight oscillations
are observed around positive states that can also be considered healthy states of
activity. In both the shoots and the roots variables, the preferred states in the
long-time projection of the model are consistent with the observed activities dur-280
ing the experiment. In Figure 2(e), we observe a slight and almost insignificant
increase in metal contaminant that might be naturally present in the environ-
ment and does not significantly affect the plant’s dynamics. Figure 2(f ) shows
a slight contaminant increase in the projected model dynamics. Overall, the
increase of the projected contaminant in both the roots and the shoots is less285
than 1 DW unit. Additionally, in Table 2, we notice that the parameters for
the control experiment differ in several orders of magnitude compared to the
rest of the experiments in which Cu contaminant is added to the plant growing
condition.
14
(a) (b)
0 1 2 3 4 5 6 7
days
1
2
3NR (Roots)
model
data interS
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
data interp
data
01234567
days
1
2
3NR (Shoots)
model
data inWHUS
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
GDWDLQWHUS
data
(c) (d)
0 2 4 6 8 10 12 14
days
1
2
3
NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
0 2 4 6 8 10 12 14
days
1
2
3
NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
1
2
3
4
5
6
7
8
9
10 Metal Accumulation
model (5)
data (5)
model (6)
GDWD (6)
0 2 4 6 8 10 12 14
days
1
2
3
4
5
6
7
8
9
10 Metal Accumulation
model (5
model (6)
model
Figure 2: N R enzyme activity, biomass content, and metal accumulation in the roots and
the shoots of C. album in a control experiment (Subject A). (a)(b)NR enzyme activity
and biomass content in the roots and the shoots of a 7-day experiment, respectively. (c)(d)
Projection of the model in a 14-days scenario. (e)(f) Modelling of plausible contaminant
dynamics in the 7-day experiment, and 14-days projection, respectively.
15
290
In Figures 3-5, we consider the model dynamics under the addition of 50 µM
of Cu to the plant growing conditions (Subject B,C, and D, respectively).
There is a good resemblance between the experimental data and the model
during the 7-day calibration and validation period in the three scenarios. In295
Figures 3(a)(b), we observe a slightly oscillatory behavior in the roots’ vari-
ables. There is a more pronounced oscillatory behavior in the shoots’ variables.
In the projected model dynamics, we observe consistent oscillatory dynamics
in the roots and a long-term toxic effect in the shoots, disrupting the shoots’
variables after day 8 (N R enzyme activity) or day 13 (biomass content). The300
concentration of the contaminant is mainly in the roots during the experiment
and the long-term projection. The model Cu concentration in the roots reaches
around 140 DW units (150 DW units in the data) at the end of the experiment
and about 220 DW units at the end of the projected time. The model C u
concentration in the shoots reaches around 90 DW units (80 DW units in the305
data) at the end of the experiment and around 140 DW units at the end of the
projected time.
In Figure 4, we observe different model projections compared to Figure 3. There
are oscillatory dynamics for both the roots’ and the shoots’ variables during the
experiment. During the projected longer time dynamics of the system, the310
variables remain near slightly decreased positive activity states. In the shoots’
variables, there is projected a slight and slow decrease in both variables. The
preferred states of each variable are consistent with the activity during the ex-
periment. Thus, in this case, there is no disruption of the activity in the shoots
but rather a slight decrease in its variables. Additionally, the contaminant is315
mainly absorbed in the roots. The Cu concentration in the roots reaches around
160 DW units (150 DW units in the data) at the end of the experiment and
around 240 DW units at the end of the projected time. The Cu concentration
in the shoots reaches around 85 DW units (85 DW units in the data) at the
16
(a) (b)
0 1 2 3 4 5 6 7
days
1
2
3
NR (Roots)
model
data interS
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
data interp
data
01234567
days
1
2
3
NR (Shoots)
model
data interp
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
data interp
data
(c) (d)
0 2 4 6 8 10 12 14
days
1
2
3NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
model
0 2 4 6 8 10 12 14
days
1
2
3NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
50
100
150
200
250 Metal Accumulation
model (5
data (5)
model (6)
GDWD (6)
0 2 4 6 8 10 12 14
days
50
100
150
200
250 Metal Accumulation
model (5
model (6)
Figure 3: NR enzyme activity and biomass content, in the roots and the shoots of C. album
in an experiment with 50 µM of Cu added to the environment (Subject B) (a)(b)N R enzyme
activity and biomass content in the roots and the shoots of C. album in a 7-day experiment,
respectively. (c)(d) Projection of the model in a 14-days scenario. (e)(f) Modelling of plausible
contaminant dynamics in the 7-day experiment, and 14-days projection, respectively.
17
end of the experiment and around 130 DW units at the end of the projected320
time.
In Figures 5(a)-(d), we observe a medium oscillatory behavior of the variables.
The long-term projections show no significant change in the variables compared
to the experiment. There is a long tendency to maintain positive “healthy”
states. The concentration of the contaminant is mainly in the roots during325
the experiment and the projected activity. The Cu concentration in the roots
reaches around 140 DW units at the end of the experiment, slightly different
from the 160 DW units in the data, and approximately 220 DW units at the
end of the projected time. The C u concentration in the shoots reaches around
80 DW units at the end of the experiment and around 130 DW units at the330
end of the projected time.
In the overall analysis of Figures 3-5, we found that the activity shown is consis-
tent with the roots’ ability to absorb contaminants facing the possibility (Fig-
ure 3) of a longer-time toxic effect in the shoots. These three scenarios support
the viability of the plant to be used in copper-contaminated areas. Moreover,335
there is a consistent metal accumulation in the three cases. At the end of the
experiment, the contaminant concentration in the roots ranges from 140 to 160
DW units and from 220 to 240 DW units for the long-time projected dynamics.
On the other hand, the contaminant concentration in the shoots ranges from 80
to 90 DW units at the end of the experiment and from 130 to 140 DW for the340
long-time projected dynamics.
Figures 6-7 consider the model dynamics under 500 µM of Cu added to the plant345
growing condition (Subjects Eand F, respectively). The resemblance between
the experimental data and the model during the 7-day calibration/validation
period is reasonable in the two cases. In Figure 6, we observe a slightly oscil-
latory behavior with a tendency to decrease over time during the experiment.
18
(a) (b)
0 1 2 3 4 5 6 7
days
1
2
3NR (Roots)
model
data interp
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
data interp
data
01234567
days
1
2
3NR (Shoots)
model
data interp
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
data interp
data
(c)
(d)
0 2 4 6 8 10 12 14
days
1
2
3NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
model
0 2 4 6 8 10 12 14
days
1
2
3NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
50
100
150
200
250 Metal Accumulation
model (5)
data (5)
model (6)
GDWD (6)
0 2 4 6 8 10 12 14
days
50
100
150
200
250 Metal Accumulation
model (5
model (6)
Figure 4: NR enzyme activity and biomass content in the roots and the shoots of C. album in
an experiment with 50 µM of Cu added to the environment (Subject C) (a)(b)N R enzyme
activity and biomass content in the roots and the shoots of C. album in a 7-day experiment,
respectively. (c)(d) Projection of the model in a 14-days scenario. (e)(f) Modelling of plausible
contaminant dynamics in the 7-day experiment, and 14-days projection, respectively.
19
(a) (b)
0
1234567
days
1
2
3NR (Roots)
model
data interp
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
data interp
data
01234567
days
1
2
3NR (Shoots)
model
data interS
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
data interp
data
(c) (d)
0 2 4 6 8 10 12 14
days
1
2
3NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
model
0 2 4 6 8 10 12 14
days
1
2
3NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
50
100
150
200
250 Metal Accumulation
model (5)
data (5)
model (6)
GDWD (6)
0 2 4 6 8 10 12 14
days
50
100
150
200
250 Metal Accumulation
model (5)
model6)
Figure 5: NR enzyme activity and biomass content in the roots and the shoots of C. album in
an experiment with 50 µM of Cu added to the environment (Subject D) (a)(b)N R enzyme
activity and biomass content in the roots and the shoots of C. album in a 7-day experiment,
respectively. (c)(d) Projection of the model in a 14-days scenario. (e)(f) Modelling of plausible
contaminant dynamics in the 7-day experiment, and 14-days projection, respectively.
20
The only exception is the shoots’ biomass content that remains near a dimin-350
ished activity state. The long-term model dynamics show a consistent toxic
effect and an activity decrease as time evolves. Such reduction is oscillatory in
the roots’ variables. In addition, the N R enzyme activity in the shoots has the
most significant activity decrease over time. The biomass content in the shoots
is projected to stay near a stable state, similar to the experiment. The metal355
concentration in the roots reaches around 360 DW units at the end of the exper-
iment, slightly different from the 330 DW units in the data and approximately
450 DW units at the end of the projected time. The metal concentration in the
shoots reaches around 200 DW units (180 DW units in the data) at the end of
the experiment and around 390 DW units at the end of the projected time.360
In Figure 7, we observe oscillatory dynamics of the data during the experiment
with a tendency to decrease over time. The projected activities in the roots’
variables show a significant decrease tending to disrupt the NR enzyme activ-
ity and the biomass content. On the other hand, the projected activities in365
the shoots’ variables show oscillatory components tending to disrupt activity
around day 14. The metal concentration in the roots reaches around 390 DW
units (360 DW units in the data) at the end of the experiment and around 480
DW units at the end of the projected time. The metal concentration in the
shoots reaches around 200 DW units (200 DW units in the data) at the end of370
the experiment and around 390 DW units at the end of the projected time.
In the analysis of Figures 6-7, our results show that the projected dynamics
are consistent with the roots’ ability to absorb contaminants, facing either a
decrease in the shoots’ variables or disruption of activities. These scenarios also375
support the viability of the model plant to be used in copper-contaminated ar-
eas. At the end of the experiment, the contaminant concentration in the roots
is around 360 DW units and ranges from 450 to 480 DW units for the projected
dynamics. On the other hand, the contaminant concentration in the shoots lies
around 200 DW units at the end of the experiment and around 390 DW units380
21
for the long-time projected dynamics. In conclusion, there is a toxic effect in
both of these scenarios due to the contaminant. However, the time scale in
which this harmful effect is present is slow, which provides a solid indication
and plausible theoretical and modelling justification of the model plant to serve
as an indicator species.385
As a final model analysis, we observe that the data and model dynamics are
consistent during the first 7-days. The ranges of metal accumulation at the end
of the experiment and the end of the long-time projections remain consistent for
the different scenarios in each of the cases considered. Moreover, when analyzing390
the parameter values in Table 2, we observe that the parameters related to
the contaminant uptake (e.g., eR,eS,r1,r2, and r3) lie in similar ranges. In
particular, the contaminant uptake rate in the roots (r1), contaminant uptake
rate in the shoots (r2), and contaminant return rate to the environment (r3)
are nearly identical in each of the cases considered (last three rows in Table 2).395
The long-time tendency of the system’s variables (i.e., the equilibrium solutions)
is dependent on the different model parameters, as is detailed in the Appendix.
Here, and motivated by our previous results, we analyze the contaminant’s toxic400
effect in the shoots activities. Figure 8 establishes a model scenario under an
in-silica experimental setting. In our results, we obtain that the long-term
tendency of the shoots’ N R enzyme activity and biomass content is determined
by three factors: the environmental factors, the amount of contaminant, and
the absorption rates in the shoots. In particular, the longer time tendency of405
the system is established by the following relationships:
NS→dS−eSN
aS
,(8)
and
22
(a) (b)
0 1 2 3 4 5 6 7
days
0.5
1
1.5
2NR (Roots)
model
data interp
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
dataLQWHUS
data
01234567
days
0.5
1
1.5
2NR (Shoots)
model
data interp
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
data interp
data
(c) (d)
0 2 4 6 8 10 12 14
days
0.5
1
1.5
2NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5
2NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
50
100
150
200
250
300
350
400
450
500 Metal Accumulation
model (5)
data (5)
model 6)
GDWD(6)
0 2 4 6 8 10 12 14
days
50
100
150
200
250
300
350
400
450
500 Metal Accumulation
model (5)
model (6)
Figure 6: NR enzyme activity and biomass content in the roots and the shoots of C. album in
an experiment with 500 µM of Cu added to the environment (Subject E) (a)(b)N R enzyme
activity and biomass content in the roots and the shoots of C. album in a 7-day experiment,
respectively. (c)(d) Projection of the model in a 14-days scenario. (e)(f) Modelling of plausible
contaminant dynamics in the 7-day experiment, and 14-days projection, respectively.
23
(a) (b)
0 1 2 3 4 5 6 7
days
0.5
1
1.5
2NR (Roots)
model
data interp
data
0 1 2 3 4 5 6 7
days
0.5
1
1.5 Biomass (Roots)
model
data inteUS
data
01234567
days
0.5
1
1.5
2NR (Shoots)
model
data interp
data
01234567
days
0.5
1
1.5 Biomass (Shoots)
model
data interS
data
(c) (d)
0 2 4 6 8 10 12 14
days
0.5
1
1.5
2NR (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Roots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5
2NR (Shoots)
model
0 2 4 6 8 10 12 14
days
0.5
1
1.5 Biomass (Shoots)
model
(e) (f)
01234567
days
50
100
150
200
250
300
350
400
450
500
Metal Accumulation
model (5)
data (5)
model (6)
GDWD6)
0 2 4 6 8 10 12 14
days
50
100
150
200
250
300
350
400
450
500
Metal Accumulation
model (5)
model (6)
Figure 7: NR enzyme activity and biomass content in the roots and the shoots of C. album in
an experiment with 500 µM of Cu added to the environment (Subject F) (a)(b)N R enzyme
activity and biomass content in the roots and the shoots of C. album in a 7-day experiment,
respectively. (c)(d) Projection of the model in 7-day experiment, and 14-days projection,
respectively.
24
BS→αS(dS−eSN)
βSaS
,(9)
where NSis the projected N R enzyme activity in the shoots and BSis the
projected biomass content in the shoots. As shown in Figure 8, strategic har-
vesting can be planned from the previous relationships, as the variables can410
provide a plausible scenario for the amount of contaminants in the environment
and a theoretically proposed uptake rate. In particular, a minimum harvesting
time can also be established, especially contemplating a framework in which the
contaminant concentration in the environment is unknown.
2 4 6 8 10 12 14
days
0
1
2
3
NR (Shoots)
0 M
50 M
300 M
500 M
2 4 6 8 10 12 14
days
0
0.5
1
1.5
2Biomass (Shoots)
0 M
50 M
300 M
500 M
Figure 8: Pro jection of the shoots’ N R enzyme activity and the biomass content under dif-
ferent concentrations of Cu.
415
4.2. Parameter Sensitivity Analysis
Sensitivity analysis serves to quantify how important each parameter is to each
model variable. Sensitivity analysis is used to measure the effects of uncertain-
ties on both the model’s input and output. This mathematical tool can help
25
create more accurate models by identifying key parameters that predominantly420
affect the model dynamics. In this case, the normalized forward sensitivity in-
dex can measure the output variable’s relative change regarding a change in a
chosen input parameter. The normalized forward sensitivity index is defined in
Arriola and Hyman (2009) as:
Sup:= lim
δp→0δu
uδp
p−1
=p
u∂u
∂p ,(10)
where uis a model variable, and pis a fixed model parameter. We are interested425
in quantifying which parameters influence the equilibrium solution most, as the
stability of such an equilibrium solution is well understood. Therefore, we now
measure the normalized sensitivity index of the biomass content and the N R
enzyme activity in both the shoots and the roots to better understand the
critical model dynamics. In Table 3, we show the non-trivial sensitivity indexes430
of our model. The most sensible parameters on the shoots’ biomass variable are
the constant environmental forcing term and the N R influence on the biomass
change rate. In particular, increasing dSor αSby 10% implies modifying the
shoots’ biomass asymptotic tendency by more than 10% (as the sensitivity is
greater than 1). In the same way, increasing dSby 10% implies modifying the435
shoots’ N R enzyme activity by more than 10%. Also, we observe that changing
the shoots’ contaminant influence rate (eS) by 10% implies modifying the shoots’
NR enzyme activity by less than 10% (as the sensitivity index is smaller than
1).
5. Discussion440
In the present study, we proposed a novel mathematical model providing a set-
ting to relate the nitrogen-assimilatory enzyme activity and the biomass content
to contaminant levels. The model parameters are calibrated and validated by
experimental data to make the model biologically more plausible. Furthermore,
different scenarios were considered to model contaminant-induced stress in the445
26
Parameter Model Equilibrium Variable Sensitivity Index
dS, αSBS1 + eSN
dS−eSN
dSNS1 + eSN
dS−eSN
dR,αRBR1
dRNR1
eSBS1−dS
dS−eSN
eSNS1−dS
dS−eSN
aS, βSBS−1
aSNS−1
aR, βRBR−1
aRNR−1
Table 3: Non-trivial sensitivity index of the model. We assume dS−eSN > 0.
model plant. In this work, we have shown, the suitability of C. album to be con-
sidered in contaminated areas through mathematical modelling and numerical
simulation. Our results show a tendency of the roots to maintain healthy levels
of activity despite the presence of high levels of Cu. This roots’ characteristic
has provided the ability of the model plant to survive in sewage sludge lagoons.450
Indeed, in Akpinar (2021), Chenopodium album L. was observed naturally es-
tablished in such an environment.
Different studies have analyzed the plausibility of C. album for remediation of
contaminated soil. Notably, in Mohan et al. (2019), the model plant removed
heavy metals from the ground through a phytoremediation process. In Bhar-455
gava et al. (2007), there was an extensive study investigating the plausibility of
Chenopodium spp. to endure six heavy metal contaminants. The abovemen-
tioned study established scientific evidence of C. album being a good copper
accumulator. It was also suggested that the metal accumulation effectiveness of
Chenopodium spp. made it a suitable choice for phytoextraction. In both of the460
previous studies, the assessment of metal uptake was made through different
experimental settings. In comparison, our results of the experimental settings
27
are supported through the evolution of the main model variables over time.
This suggests that the ability of C. album, shown in the experimental data, to
concentrate heavy metals in the roots and the shoots can also be maintained465
for longer times.
Tozser et al. (2019) analyzed metal uptakes in C. album. In this study, it was
suggested that to achieve efficiency in the remediation process, it is necessary
to remove the plant organs lying above the ground once the contaminant is
accumulated. Our model could potentially provide a theoretical strategy for470
such a removal process. Indeed, the removal process could be established as an
optimization problem based on an estimated amount of contaminant in the soil.
Different studies have also assessed the ability of C. album to be suitable for
phytoextraction of lead (Alipour et al., 2015), phytoextraction of different met-
als (Gupta and Sinha, 2007), and phytoremediation of cadmium from the soil475
(Zulfiqar et al., 2012), and they have obtained promising results. Additionally,
we observe that the relevant contaminants previously mentioned have similar
equivalence (Cu2+,C d2+, among others). Therefore, mathematical modelling
considering different contaminants with similar equivalence can provide infor-
mation about the plausible long-term plant dynamics under various heavy metal480
contaminants. In this case, our model provided theoretical evidence of the ca-
pability of the model plant to maintain functional dynamics in the roots under
Cu-induced stress.
Our results show that increasing contaminant concentrations mainly affect the
shoot activities, and toxic effects can be observed. Another important obser-485
vation of the model is that there is root capability to absorb the contaminant
initially. However, as time increases, the shoots have a slower tendency to absorb
contaminants. As is shown in Figure 8, the slower trend of the shoots to assim-
ilate contaminants is severely affected by the concentration in the system. In
particular, as contaminant concentration increments, the activity in the shoots490
tends to be disrupted. Thus, being exposed to higher contaminant levels implies
a higher absorption of contaminant and quicker disruption of healthy dynam-
ics. Therefore, under the scenario shown in Figure 8, our model can potentially
28
establish a harvesting strategy dependent on the environmental dynamics.
A mathematical model is always improvable. Thus, future work can be directed495
to improve some of the model’s limitations. A main future goal is to relate
the shoots and the roots dynamics, here considered independent, by coupling
the corresponding variables. However, this can potentially increase the model’s
complexity and ability to relate to experimental observations. Therefore, this
modification needs to be carefully established to combine a more realistic model500
without losing the model’s tractability and ability to relate to the experimental
data. Also, further inclusion of plant features such as chlorophyll contents and
total proteins content needs to be addressed in future modelling. Our sensitiv-
ity analysis also shows that the model is sensitive to modifying the parameters
dS, αS,dR, and αR. Therefore, even though our model can correctly capture505
the contaminant dynamics and its accumulation in the shoots and roots, our
simple model needs to be improved to consider more realistic complex internal
processes in the plant’s dynamics.
Our mathematical model can motivate basic experimental settings to analyze
the effect of the contaminant on different plant species used in phytoremedia-510
tion. Furthermore, as shown in this work, our model can potentially predict
the NR enzyme activity and the biomass content in the roots and shoots and
propose a plausible scenario for the contaminant concentration.
6. Conclusions
In this work, we established a mathematical model to support the suitability515
of the model plant (Chenopodium album L.) to be used in contaminated ar-
eas. Here, we provided a model for the nitrogen-assimilatory enzyme activity,
the biomass content, and the contaminant concentration in the plant. As far
as the authors know there is no similar existent model. Our numerical model
simulation results provided qualitatively similar activities compared to the ex-520
perimental data. Indeed, the main goal of the model was to give a qualitative
description of the main variables and long-term projections of their dynamics.
29
In these long-term projections, we observed a consistent tendency of maintain-
ing normal or diminished activities of the main variables, depending on the
contaminant-induced stress. We observed a plausible tendency to disrupt reg-525
ular shoots’ dynamics, whereas a consistent trend to maintain normal roots’
dynamics. Moreover, we observed a consistent and longer-term tendency to
absorb more significant amounts of contaminants in the roots. Even though
our main goal was to provide a qualitative description of the main variables,
our results can potentially be used to describe quantitative features of the main530
variables, e.g., the amount of metal accumulation in the plant, due to their abil-
ity to resemble the experimental data. Therefore, this model provides scientific
progress related to quantifiable plant dynamics related to contaminant accumu-
lation and provides the basis for further experimental observations.
Our theoretical results provided expressions for the long-term dynamics in the535
shoots variables. This complements the analysis developed in the manuscript,
and it is a straightforward way to analyze the long-term dynamics of the sys-
tem. Also, our sensitivity analysis provided information about some of the
model’s limitations and further directions for the mathematical improvement of
the model.540
Our mathematical model complemented experimental studies performed under
copper-induced stress of Chenopodium album and provided the foundation of
main variables to be analyzed in future experimental settings. In this way, both
experimental studies and mathematical modelling can provide complementary
information that supports biological hypotheses. Future research opportunities545
can be directed towards incorporating more biological variables of the model,
increasing model complexity, and comparing model performance under different
heavy metal contaminants.
There is a need for continuing efforts to join both theoretical and experimental
designs to establish further scientific advances to address the problem of heavy550
metal contamination in the environment. Multidisciplinary research teams need
to continue this direction as it has successfully shown that better results can be
achieved than solely theoretical or experimental approaches.
30
7. Appendix
This Appendix establishes the mathematical model used to set the parameter555
optimization problem. When considering the constant-coefficient matrix of the
model (6), we obtain a unique equilibrium point determined by:
NR, TR, BR, NS, TS, BS, ME, MR, MS=
dR
aR
,0,αRdR
aRβR
,dS−eSN
aS
,0,αS(dS−eSN)
βSaS
,0,0, N .
(11)
The system’s eigenvalues consist of a zero eigenvalue and 8 real eigenvalues
with a negative real part. Finally, we note that the last three equations of the560
model (6) can be decoupled obtaining the system:
dME
dt =−(r1+r2)ME+r3MR
dMR
dt =r1ME−r3MR
dMS
dt =r2ME.
(12)
The previous linear system can be explicitly solved by employing the eigenvalues
and the eigenvectors of the corresponding constant coefficient matrix. Therefore,
it is possible to obtain explicit expressions for the contaminant level. With
the previous information, we can update the model to simplify the parameter565
inference to obtain:
dNR
dt =TR
dTR
dt =dR−bRTR−aRNR−eRMR
dBR
dt =αRNR−βRBR
dNS
dt =TS
dTS
dt =dS−bSTS−aSNS−eSMS
dBS
dt =αSNS−βSBS,
(13)
31
where ME(t), MS(t), and MR(t) can be explicitly described. In this way, it
is possible to reduce the number of differential equations and remain with 13
parameters to be determined.
570
By considering the fixed point of the new system and its stability, we provide
the asymptotic tendency of the plants’ shoots and roots activities:
NR→dR
aR
,(14)
BR→αRdR
βRaR
,(15)
NS→dS−eSN
aS
,(16)
and
BS→αS(dS−eSN)
βSaS
.(17)
As the contaminant increases, the shoots tend to decrease their variables. The
rise in contaminants exerts a toxic effect, determined by the contaminants ab-575
sorption rate. Nevertheless, we remark that this model’s asymptotic limits are
only helpful in the case of positive variable values. Therefore, detailed experi-
mental data needs to motivate the choice of parameters for this analysis.
Competing Interests
The authors declare no competing interests.580
Acknowledgements
LRGR was funded by SIP-IPN 2021-1285 and SIP-IPN 2022-1416.
32
Author contributions
L.R. Gonz´alez-Ram´ırez: Conceptualization, methodology, investigation, for-
mal analysis, writing original draft and review, software, visualization. D.585
Ala¸cam: Supervision, writing original draft. A. Akpinar: Methodology,
investigation, data curation, resources, validation, writing original draft. All
authors have read and agreed to the published version of the manuscript.
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