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 [

]. 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 [

]) and disease

management (for example, organophosphate insecticides which kill mosquitoes to

control spread of diseases such as malaria and dengue [

]). 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

[

]. 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 [

], but often these models focus on root exposure (see, for example,

[

]) 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.

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 [

]. 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 [

]. 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 [

]. For example, an accelerator AJ [

] will increase the rate at which

the AI can move across the leaf cuticle [

] by increasing the ﬂexibility or ﬂuidity of

the waxes and cutin [

]. Another common AJ is a surfactant [

] 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

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

[

]. 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 [

]

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 =−λAAPA(t)−κA,1

M1(0, t)

A,(2.1a)

∂M1(x, t)

∂t =DP

∂2M1(x, t)

∂x2,(2.1b)

d(VBPB)(t)

dt =λBAκB,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

= 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 =λAAPA(t)−κA,1

M1(0, t)

A,(2.2d)

−DP

∂M1(L, t)

∂x =λBAκB,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,01 + αM1

σ+M1,

where

DQ,0

and

are positive constants as deﬁned in Table 1. The shape of this

function is shown in Figure 3.

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

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

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