Mathematical modelling of adjuvant-enhanced active ingredient leaf uptake of pesticides

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MATHEMATICAL MODELLING OF ADJUVANT-ENHANCED
ACTIVE INGREDIENT LEAF UPTAKE OF PESTICIDES∗
JENNY DELOS REYES†, TONY SHARDLOW‡, M. BEGO ˜
NA DELGADO-CHARRO§,
STEVEN WEBB,AND K. A. JANE WHITE‡
Abstract.
The global importance of effective and affordable pesticides to optimise crop yield
and to support health of our growing population cannot be understated. But to develop new products
or refine existing ones in response to climate and environmental changes is both time-intensive and
expensive which is why the agrochemical industry is increasingly interested in using mechanistic
models as part of their formulation development toolbox. In this work, we develop such a model to
describe uptake of pesticide spray droplets across the leaf surface. We simplify the leaf structure by
identifying the outer cuticle as the main barrier to uptake; the result is a novel, hybrid model in which
two well-mixed compartments are separated by a membrane in which we describe the spatio-temporal
distribution of the pesticide. This leads to a boundary value partial differential equation problem
coupled to a pair of ordinary differential equation systems which we solve numerically. We also
simplify the pesticide formulation into two key components: the Active Ingredient which produces the
desired effect of the pesticide and an Adjuvant which is present in the formulation to facilitate effective
absorption of the Active Ingredient into the leaf. This approach gives rise to concentration-dependent
diffusion. We take an intuitive approach to parameter estimation using a small experimental data set
and subsequently demonstrate the importance of the concentration-dependent diffusion in replicating
the data. Finally, we demonstrate the need for further work to identify how the physicochemical
properties of pesticides affect flow into and across the leaf surface.
Key words.
pesticide, hybrid ODE-PDE model, parameter estimation, concentration-dependent
diffusion, physico-chemical properties
MSC codes. 9210, 92F05
1. Introduction.
The importance of effective and affordable agrochemicals world-
wide cannot be understated [
38
]. They are used globally to optimise crop yield in
a number of different ways including growth enhancement (for example, herbicides
that kill unwanted plants to eliminate competition for resources [
10
,
1
]) and disease
management (for example, organophosphate insecticides which kill mosquitoes to
control spread of diseases such as malaria and dengue [
27
]). Traditionally the process
of developing a new product is lengthy and expensive, and this has helped to strengthen
recent interest in adding mathematical models to the product development toolbox
[
17
]. The model which we present here is our contribution and we focus on the uptake
of pesticides through the leaf surface.
There is a small literature on mechanistic models for pesticide uptake in plants as
summarised in [
9
], but often these models focus on root exposure (see, for example,
[
36
,
11
]) and not direct leaf surface contact with the pesticide spray. An uptake model
that incorporates foliar exposure and whole plant allocation of absorbed chemical has
been presented in [28,29], and was used here as a basis for model development.
One challenge in modelling uptake across the leaf surface is to determine the
appropriate simplifying assumptions about the leaf structure (see Figure 1) which
∗Funding:
This work was funded by Engineering and Physical Sciences Research Council grant
EP/S515279/1 and by Syngenta UK Ltd. grant TK0448301.
†
Corresponding author. Department of Mathematical Sciences, University of Bath, Bath, UK,
BA2 7AY (jdr47@bath.ac.uk).
‡Department of Mathematical Sciences, University of Bath, Bath, UK, BA2 7AY.
§Department of Life Sciences, University of Bath, Bath, UK, BA2 7AY.
Product Safety, Syngenta, Jealott’s Hill International Research Centre, Bracknell, UK, RG42
6EY.
1
arXiv:2210.11205v1 [math.DS] 20 Oct 2022
2J. DELOS REYES, T. SHARDLOW, M.B. DELGADO-CHARRO, S. WEBB, K.A.J. WHITE
consists of many layers and which is also laterally heterogeneous [
28
,
30
,
19
]. One
layer that is clearly identified as important in the uptake process is the uppermost
layer, known as the cuticle. This layer is non-cellular, made of cutin which is a
waxy substance [
14
,
30
]. The cuticle acts as the main barrier to prevent water loss
or cuticular transpiration, and it protects the leaf from any external threat such as
chemical attacks and ultraviolet radiation [23].
Fig. 1: Simplified diagram of the leaf structure.
As with the leaf, the composition of pesticides are highly complex. Each formula-
tion will necessarily include an
Active Ingredient (AI)
because this is the compound
that actually works against pests by controlling, killing, or repelling them. Beyond
that, formulations include a wide range of other compounds each of which performs
some function. One of the more common compounds in a formulation is known more
generally as an
Adjuvant (AJ)
which is added to facilitate the absorption of AI into
the leaf, however the exact mechanisms are extremely complex, and only partially
understood [
37
,
1
]. For example, an accelerator AJ [
31
] will increase the rate at which
the AI can move across the leaf cuticle [
30
] by increasing the flexibility or fluidity of
the waxes and cutin [
32
]. Another common AJ is a surfactant [
34
] which acts on the
contact between pesticide droplet and leaf surface in order to optimise delivery of the
AI.
With leaf structure and pesticide composition in mind, in the following Section 2 we
formulate our model and elucidate our simple approach to model parameter estimation
using a small data set. Section 3 presents our results which consist of numerical
solutions of the model system. These numerics involve solving a boundary value partial
differential equation (PDE). Our results highlight the importance of the AJ in creating
predictions that are consistent with experimental data. Finally in Section 4 we discuss
limitations of our model and identify future work to develop more robust empirical
relations between key model processes and the underlying physico-chemical properties
of the pesticide components.
2. Model formulation.
In our model, we assume a simplified leaf anatomy
comprising two layers, namely the wax/cuticle and the leaf tissue, together with a
pesticide droplet which instantaneously settles to a stable configuration on the leaf
surface. We consider the case of no evaporation, which means that all volumes and
contact areas between layers and the pesticide droplet remain constant throughout.
MATHEMATICAL MODELLING OF PESTICIDE LEAF UPTAKE 3
For simplicity we assume that the droplet shape can be represented by a hemisphere
and that each droplet is identical and spatially segregated from all other droplets.
We assume that the pesticide moves from the droplet across the leaf cuticle into the
leaf tissue in the transverse direction only so that our problem becomes spatially
one-dimensional. This approach, in which lateral movement within the cuticle is
ignored, has been used widely in the literature for flow across biological membranes
(see, for example, [5,15,12,35,16]).
As for leaf anatomy, we use a simplified description of the pesticide formulation
using two components - AI and AJ. The simplification lies in our decision to combine
all compounds in the formulation which act to enhance uptake of AI across the leaf
cuticle into a single variable AJ. Figure 2 summarises our model structure.
Fig. 2: Schematic diagram of the model system at time
t
where state variables are
defined in Table 1.
The leaf cuticle acts as a barrier for the plant and so, as with other biological
barriers such as human skin, flow of solvents across the barrier is naturally slow
[
30
,
18
]. This property determines our use of a spatially explicit system within
the leaf cuticle. By contrast, in the droplet and the leaf tissue, we assume that
both AI and AJ are well-mixed and consequently we describe them using ordinary
differential rate equations. Furthermore, we assume that the droplet and leaf tissue
have similar aqueous properties (relevant in our parameter estimation), that there is
no crystallisation or photodegredation in the droplet and no metabolism within the
leaf tissue and hence loss of AI and AJ from our system only occurs from the leaf
tissue as the compounds move elsewhere in the leaf structure.
We describe flow between the three model compartments with partitioning [
21
]
which corresponds to the rate of flow being proportional to a weighted difference
between current concentrations in the neighbouring compartments. The weightings
are determined by the physico-chemical properties of the compound such that, flow
between compartments ceases when the ratio of concentrations in each compartment
equals to the partitioning coefficient.
As mentioned above, in the leaf cuticle we use a one-dimensional spatio-temporal
model. We model flow of the AJ in the cuticle using a simple diffusion process with
a constant diffusion coefficient. We also use a diffusion process to describe the flow
of AI across the leaf cuticle, but in this case we assume that the diffusion coefficient
4J. DELOS REYES, T. SHARDLOW, M.B. DELGADO-CHARRO, S. WEBB, K.A.J. WHITE
depends on the local concentration of the AJ. In particular, we choose a saturating
function for the diffusion coefficient as described below.
Using the state variables presented in Figure 2 and defined in Table 1 and noting
that
M1(x, t) = AP1(x, t)
our model system for the adjuvant is given as:
d(VAPA)(t)
dt =−λAAPA(t)−κA,1
M1(0, t)
A,(2.1a)
∂M1(x, t)
∂t =DP
∂2M1(x, t)
∂x2,(2.1b)
d(VBPB)(t)
dt =λBAκB,1
M1(L, t)
A−PB(t)−βVBPB(t).(2.1c)
where all model parameters, defined in Table 1 are positive. To fully specify the
system, and to ensure no loss of material across the membrane boundaries
x
= 0 and
x=L, we impose initial and boundary conditions:
PA(0) = P0
A,(2.2a)
M1(x, 0) = 0,(2.2b)
PB(0) = 0,(2.2c)
−DP
∂M1(0, t)
∂x =λAAPA(t)−κA,1
M1(0, t)
A,(2.2d)
−DP
∂M1(L, t)
∂x =λBAκB,1
M1(L, t)
A−PB(t).(2.2e)
The model system for AI follows a similar structure except that the diffusion
coefficient for the AI within the cuticle is assumed to be a saturating function of
the local AJ concentration. This represents the facilitating role that the AJ plays in
moving the AI into the leaf tissue.
In the absence of any data, we assume that the concentration-dependent diffusion
coefficient satisfies the parsimonious conditions:
•In the absence of AJ, the AI will still diffuse but more slowly.
•
The effect of AJ is saturating such that at high AJ concentration, the impact
on diffusion of AI is limited.
•
The impact of AJ on diffusion of AI increases monotonically with AJ concen-
tration.
An example of such a function which we use in our numerical simulations of the system
is given by the algebraic expression
(2.3) DQ(M1) = DQ,01 + αM1
σ+M1,
where
DQ,0
,
α
and
σ
are positive constants as defined in Table 1. The shape of this
function is shown in Figure 3.
摘要：

MATHEMATICALMODELLINGOFADJUVANT-ENHANCEDACTIVEINGREDIENTLEAFUPTAKEOFPESTICIDESJENNYDELOSREYESy,TONYSHARDLOWz,M.BEGO~NADELGADO-CHARROx,STEVENWEBB{,ANDK.A.JANEWHITEzAbstract.Theglobalimportanceofe ectiveanda ordablepesticidestooptimisecropyieldandtosupporthealthofourgrowingpopulationcannotbeunderstat...

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