Coupling kinetic models and advection-diffusion equations. 1. Framework development and application to sucrose translocation and metabolism in ...
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
in silico Plants Vol. 3, No. 1, pp. 1–18
doi:10.1093/insilicoplants/diab013
Advance Access publication 25 March 2021
Original Article
Coupling kinetic models and advection–diffusion
equations. 1. Framework development and
application to sucrose translocation and metabolism
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
in sugarcane
Lafras Uys, Jan-Hendrik S. Hofmeyr, and Johann M. Rohwer*,
Laboratory for Molecular Systems Biology, Department of Biochemistry, Stellenbosch University, Private Bag X1, Matieland, Stellenbosch 7602,
Western Cape, South Africa
*Corresponding author’s e-mail address: jr@sun.ac.za
This paper is dedicated to the memory of Prof Prieur du Plessis, whose help in conceiving and setting up this analysis has been invaluable.
Handling Editor: Steve Long
Citation: Uys L, Hofmeyr J-HS, Rohwer JM. 2021. Coupling kinetic models and advection–diffusion equations. 1. Framework development and application to
sucrose translocation and metabolism in sugarcane. In Silico Plants 2021: diab013; doi: 10.1093/insilicoplants/diab013
A B ST R A CT
The sugarcane stalk, besides being the main structural component of the plant, is also the major storage organ for car-
bohydrates. Previous studies have modelled the sucrose accumulation pathway in the internodal storage parenchyma of
sugarcane using kinetic models cast as systems of ordinary differential equations. To address the shortcomings of these
models, which did not include subcellular compartmentation or spatial information, the present study extends the original
models within an advection–diffusion–reaction framework, requiring the use of partial differential equations to model
sucrose metabolism coupled to phloem translocation. We propose a kinetic model of a coupled reaction network where
species can be involved in chemical reactions and/or be transported over long distances in a fluid medium by advection
or diffusion. Darcy’s law is used to model fluid flow and allows a simplified, phenomenological approach to be applied
to translocation in the phloem. Similarly, generic reversible Hill equations are used to model biochemical reaction rates.
Numerical solutions to this formulation are demonstrated with time-course analysis of a simplified model of sucrose accu-
mulation. The model shows sucrose accumulation in the vacuoles of stalk parenchyma cells, and is moreover able to dem-
onstrate the upregulation of photosynthesis in response to a change in sink demand. The model presented is able to capture
the spatio-temporal evolution of the system from a set of initial conditions by combining phloem flow, diffusion, transport
of metabolites between compartments and biochemical enzyme-catalysed reactions in a rigorous, quantitative framework
that can form the basis for future modelling and experimental design.
K E Y W O R D S : Advection–diffusion–reaction; partial differential equations; phloem flow; simulation; sugarcane.
1. INTRODUCTION 2000a; Walsh et al. 2005). Sucrose is the most abundant soluble carbo-
Sugarcane is part of the family of grasses (Moore and Maretzki 1996). hydrate found in sugarcane and is actively accumulated to levels much
Unlike some of the other grass crops, like maize, sorghum and barley, higher than in most other plants (Bull and Glasziou 1963; Moore and
it is the stalk (also called a culm) which acts as the primary sink for Maretzki 1996).
carbohydrates produced during photosynthesis (Komor 2000b). The A number of factors influence the amount and rate of sucrose
stalk is divided into alternating nodes and internodes, with a single accumulation. Environmental conditions such as temperature, sun-
leaf attached to a node. Sucrose is primarily synthesized in the leaves light (or rather incident radiation), rainfall and soil types play a role
and loaded into the phloem, where it is transported up or down the (van Dillewijn 1952; Alexander 1973). Agricultural practices such
stalk and unloaded in the storage parenchyma (van Bel 1993; Komor as fertilization and chemical ripening can speed up accumulation
© The Author(s) 2021. Published by Oxford University Press on behalf of the Annals of Botany Company.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted
• 1
reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
2 • Uys et al.
(van Dillewijn 1952; Alexander 1973). Biochemical factors such as Fourier Amplitude Sensitivity Test (FAST) are introduced in the
genetics (Ming et al. 2001), futile cycling of sucrose (Wendler et al. accompanying paper (Uys et al. 2021).
1990; Komor 1994; Whittaker and Botha 1997; Zhu et al. 1997;
Rohwer and Botha 2001) and membrane transporters will limit the 2. M ET H O D S
maximum accumulation possible (Lemoine 2000; Rae et al. 2005b; 2.1 Modelling ADR systems
Reinders et al. 2006). We first review briefly the existing approaches for modelling phloem
Kinetic modelling is a useful tool for analysing the control and flow, and then integrate them with biochemical reactions to formulate
regulation of metabolic pathways (Rohwer 2012), and the application the ADR framework.
of systems biological approaches in the context of sugarcane physiol-
ogy have been reviewed in Rohwer and Uys (2014). To understand 2.1.1 Models of phloem flow. Sap flow in plants involves both the
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
better the factors affecting and controlling sucrose accumulation in the xylem and the phloem and has been studied extensively for many dec-
sugarcane culm storage parenchyma, Rohwer and Botha (2001) have ades; a number of models have been formulated and a recent compre-
proposed a kinetic model of the most important biochemical reactions hensive review ( Jensen et al. 2016) focuses on the biophysical aspects.
involved. A key result from this model is the important role played The phloem transports carbohydrates (mainly sucrose) from the sites
by the futile cycling of sucrose (Whittaker and Botha 1997), i.e. the of photosynthesis to the sink tissues in the plant, and comprises a
process by which sucrose is broken down and resynthesized, which is whole complex of cells. The sieve elements are arrayed end-to-end,
known to expend metabolic energy in the hexokinase and fructokinase separated from each other by a sieve plate, forming the sieve tube. The
reactions. The model showed that factors leading to a decrease in futile sieve tube in turn is surrounded by companion cells, one or more to
cycling cause a concomitant increase in sucrose accumulation. A vari- each sieve element (van Bel et al. 2002; van Bel 2003). Thompson
ation of the model was subsequently used by Schäfer et al. (2004) to (2006) described phloem with three analogies.
study the kinetics various isoforms of the enzyme sucrose synthase.
Bosch (2005) added the branch towards trehalose synthesis to the ‘A sieve tube is like dialysis tubing’
original model and concluded that partitioning to trehalose does not There is constant movement of solutes and water between the phloem
significantly impact sucrose accumulation. Uys et al. (2007) extended and the surrounding tissue. Turgor pressure is regulated through
the original model of Rohwer and Botha by explicitly modelling the changes in solute concentrations. If phloem loading and unloading
sucrose synthase and fructokinase isoforms, as well as adding the were only to occur at the extremities of a phloem tube, pressure gradi-
phosphofructokinase, pyrophosphate-dependent phosphofructoki- ents would be difficult to control as local changes in pressure gradients
nase and uridine diphosphate (UDP)-glucose dehydrogenase steps. In become difficult to induce.
an attempt to model physiological changes during culm maturation, a
steady state was calculated for eight different sets of enzyme maximal ‘A sieve tube is like a capillary’
activity data, representing internodes 3–10. Importantly, all the above Phloem flow is dominated by shear forces rather than inertial forces; in
models only focused on the reactions around sucrose and only in the other words, viscosity trumps momentum. As a consequence, pressure
cytoplasmic space (symplast) to study the effect of sucrose break- gradients develop as a result of energy dissipation. If viscosity increased
down and synthesis; they did not explicitly include other subcellular (i.e. viscous forces started to dominate), the flow rate of solute would
compartments. decrease. In this context, Thompson and Holbrook (2004) studied ‘infor-
Overall, the model in Uys et al. (2007) showed higher sucrose mation transmission’ through the phloem. Under certain conditions it is
concentrations than that in Rohwer and Botha (2001). However, possible for a local change in solute concentration or pressure to propa-
these levels still did not come close to measured concentrations, most gate through the phloem faster than the actual velocity of the phloem sap
probably because the vacuole, apoplast and phloem were not explicitly (Thompson 2005, 2006). The implication is that message passing along
included. To address these shortcomings and develop a more realistic the phloem is possible in a reasonable amount of time.
model of sucrose accumulation, we present in this paper a framework
for combining phloem flow and reaction networks into a single model. ‘A sieve tube is like a full bathtub’
Explicitly modelling the phloem compartment requires an entirely
A change in the solute concentration at a local point along the phloem
new approach, because now geometry and sap flow become important
translates into a flux of solute, towards or away, from the surrounding
and the reactions can no longer be assumed to occur in a ‘well-stirred’
sap. This ripple effect propagates along the phloem tube and becomes
homogeneous space. The processes involved in sucrose accumulation
more diffuse as it moves further away from the source of the perturba-
can be collectively referred to as advection–diffusion–reaction (ADR)
tion. The absolute magnitude of the flux is proportional to the solute
behaviour, and as shown below, this can be modelled as a set of partial
concentration and also the gradient of the solute concentration.
differential equations (PDEs).
Bieleski (2000) points out four features of sugarcane that separate
Since the ADR framework uses PDEs, classical methods for deter-
it from other plants.
mining and quantifying the effect of a parameter change on system
behaviour (such as metabolic control analysis; Kacser and Burns 1973; 1. The plant can be seen as a row of sinks all connected by the
Heinrich and Rapoport 1974) can no longer be applied. To overcome phloem.
this, and to provide two solutions for performing a parameter-sensi- 2. Sinks with widely differing sugar content all ‘feed’ off the same
tivity analysis on such systems, the Morris sampling method and the phloem strands.
Coupling kinetic models and advection–diffusion equations 1 • 3
3. Phloem unloading occurs along the entire length of the Table 1. Notation, symbols and units
phloem tube. This is in contrast to other plants where the
phloem has a well-defined ‘terminus’ in the roots. Symbol Description SI unit
4. Any point along the phloem can act as a loading or unloading site. σ set of species
The Münch hypothesis (Münch 1926, 1927, 1930) (Knoblauch and
χ set of compartments
Peters 2017, provide an historical overview) is the premise for most
ρ set of reactions
modelling of fluid flow in xylem and phloem (Daudet et al. 2002;
τ set of transporters
Lampinen and Noponen 2003; Hölttä et al. 2006). Fluid flow is
Θ control volume m3
O control surface m2
driven by a pressure gradient generated by an osmotic potential; the
S amount of species S mol
effect has been confirmed by Gould et al. (2005). Thompson and
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Holbrook (2003a) have developed a mass-balanced non-steady-state s concentration of species S mol m−3
model of phloem flow and in Appendix A of their paper also provide s vector of species concentrations
ci indexed signed stoichiometric
a concise review of phloem modelling based on Münch’s hypothesis.
De Schepper et al. (2013) have reviewed mechanisms and controls of coefficient
phloem transport, and more recently, Minchin and Lacointe (2017) N stoichiometric matrix
t time s
have formulated a model to analyse the effects of phloem unloading and
reloading on flows between source and sink. D diffusion coefficient m2 s−1
In a series of follow-up articles, Thompson and Holbrook (2003b, D matrix of diffusion coefficients
u phase average velocity m s−1
2004) used dimensional analysis to study their approach to phloem
U matrix of phase average velocities
modelling. The assumption of equilibrium between phloem sap and
κ specific conductivity m2
apoplast water potential was shown to be plausible (Thompson and
µ viscosity kg m−1 s−1
Holbrook 2003b). This is important for the model formulation pre-
(Pa s)
sented later, since passive water flux between the phloem and apo-
P pressure kg m−1 s−2
plast need not be accounted for (Thompson 2006). Furthermore, a
(Pa)
relatively small pressure drop over a small distance would allow more
R universal gas constant J K−1 mol−1
accurate control of solute loading.
T absolute temperature K
el unit vector in dimension l
2.1.2 Transport phenomena. Advection–diffusion–reaction models
∇ gradient
are examples of a class of problems called transport phenomena (Beek
∇• divergence
et al. 1999). The mathematics describing the movement of a measur-
v reaction rate mol m−3 s−1
able quantity turns out to be similar for a large range of phenomena.
v vector of reaction rates
Central to any transport phenomenon is the principle of conservation.
Vf limiting rate of reaction (maximal mol m−3 s−1
For example, the diffusion of heat or the diffusion of a solute through
velocity)
a liquid would both depend on the divergence of the gradient together
S0.5 half-saturation constant for species S mol m−3
with some, possibly constant, coefficient. Mathematically the expres-
Keq equilibrium constant
sions are identical even though two different physical properties are
Γ mass-action ratio
being modelled. The differences are the actual values and units of the
variables and parameters, together with the boundary and initial con-
ditions imposed. All of these can be measured experimentally.
Let i ∈ σ index a set of species, j ∈ χ index a set of compartments,
k ∈ ρ ∪ τ index a set of reactions and transporters, and l ∈ {1, 2, 3}
2.1.3 The ADR framework. For a summary of the notation used, refer index a set of Cartesian coordinate axes. Using the following shorthand,
to Table 1. Suppose a molecule, S , with concentration, s, is found in a
small control volume, Θ, bounded by a surface, O (Fig. 1). The number ∂
∂t ≡ ,
of S in Θ can change if the following occurs: (1)∂t
1. S is involved in a set of enzyme-catalysed reactions, ρ , found in the following set of PDEs desribes a system in which a species, sij , may
Θ, with each reaction having signed stoichiometric coefficients, be advected, diffused, transformed into another type of species or be
ci, and rate, v. moved into a neighbouring compartment,
2. S crosses O because of
(a) convection, e.g. advection by a velocity vector field, ∂t sij + ∂l (uijl sij ) + el · ∂l Di ( el · ∂l sij ) = cik vk ,
l l l k ij
(b) diffusion due to a concentration gradient, with some
coefficient, D, or (2)
(c) facilitated or active transport due to a set of transporters, where u is the phase average velocity and e is a unit vector.
Ä ä
∂ ∂ ∂
τ , with each transport step having signed stoichiometric Define the del operator ∇ ≡ e1 ∂x + e2 ∂y + e3 ∂z = ∂x , ∂y , ∂z .
coefficients, ci, and a rate, v, as in the case of a reaction. For a scalar-valued function f (x, y, z) of multiple variables, ∇f refers4 • Uys et al.
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Figure 1. Possible changes in the number of a molecule, S, in a small control volume.
to the gradient, while for a vector field u(x, y, z), ∇ • u refers to the made to Equation (3) to model translocation of photoassimilate in the
divergence. Further, let N be a stoichiometric matrix, v a vector of phloem.
reaction rates and s a vector of species concentrations. Equation (2)
can be written in matrix form as 2.1.4 ‘Solute-driven advection’. Long-distance photoassimilate
transport in plants occurs via the phloem (Bieleski 2000, van Bel
ṡ + ∇ • (Us) + ∇ • (D∇s)= Nv, 2003). All further modelling in this work will only consider a single
(3)
solute (i.e. sucrose) in the phloem; therefore, we will only consider
where the general rate vector, v = v(s, p, r, t) is a function of species Equation (3) for one variable. The right-hand side of Equation (3) can
concentrations, a parameter vector ( p), a position vector (r) and time. also be reduced to phloem loading and unloading steps, which yields
The vector v can be partitioned into reaction equations, ρ , and cross-
membrane transport equations, τ , while s can be partitioned over χ ṡ + ∇ • ( us) + ∇ • (D∇s) = vl (z) − vu (z).
into sets of species grouped by compartment. Furthermore, U = U(s) (5)
is a matrix with compartment-specific phase average velocities on the The phase average velocity, u , can be obtained from Darcy’s law (Bear
diagonal. Similarly, D = D(s) is a matrix with diffusion coefficients on 1972),
the diagonal. Each species has a unique diffusion coefficient.
u = −κ/µ · ∇P,
Note that Equation (3) is a specific case of a more general conser- (6)
vation equation, which for an arbitrary variable, ϕ , can be written as
follows: where κ is the hydraulic conductivity (a measurable quantity assumed
to be constant, Thompson and Holbrook 2003b), µ is the viscosity, P
∂t ϕ + ∇f (t, r, ϕ, ∇ϕ) = g(t, r, ϕ), is pressure and ∇P is the pressure gradient. By substituting Equation
(4)
(6) into the second term of Equation (5) the following is obtained,
where t is time and ∇ is the gradient operator, f and g are functions of
time, position (r ), ϕ and the gradient of ϕ (∇ϕ); f and g are, respec- ∇ • ( us) = ∇ • (s · (−κ/µ · ∇P))
tively, called the flux and source. (7) = −κ/µ · ∇ • (s∇P).
To solve Equation (3) for s the following need to be specified: the
equations that govern reaction rates and phase average velocities, dif- This formulation casts advection in terms of a pressure gradient, which
fusion coefficients, the geometry and subsequent domain on which can in turn be solved by using the van’t Hoff equation for dilute solu-
the variables are defined, the initial conditions and possible boundary tions (Atkins and de Paula 2002),
conditions that may exist. Given sufficient initial conditions the time-
dependent behaviour of the system can be modelled. P = Π + P0 = RTs + P0 ,
As it stands, Equation (3) is a very broad framework in which a
model needs to fit. Not all compartments will allow advection to ∴ ∇P = RT · ∇s + 0,
occur, some species may diffuse quickly through a compartment to (8)
form a homogeneous concentration field, other compartments will where Π is the osmotic pressure and P0 is the surrounding pressure
allow some combination of advection, diffusion and reaction to occur. from the apoplast. Substitution of Equation (8) into Equation (7)
The next section deals specifically with the simplifications that can be gives the following relation,Coupling kinetic models and advection–diffusion equations 1 • 5
Å ã
∇ • ( us) = −κ/µ · ∇ • (s · (RT · ∇s)) v Γ s
= 1− · .
(9) = −κ/µ · RT · ∇ • (s · ∇s). V
(14)
f K eq 1 + s+p
The conservation equation is therefore of the following form, For the dynamic system as defined in Equation (3), enzymes are also
subject to movement by advection–diffusion. This means that enzyme
ṡ − κ/µ · RT · ∇ • (s · ∇s) + ∇ • (D · ∇s) = vl (z) − vu (z). concentrations (call them e ) will change with position and therefore
their limiting rates (maximal activities) will also change. Nothing pro-
Furthermore, the contribution of the diffusion term to phloem trans- hibits treating the enzyme concentrations as state variables as well
location is typically two orders of magnitude smaller than the con- with appropriate equations for enzyme synthesis and degradation in
tribution of the advective term (Henton et al. 2002; Thompson and v . However, in this work it is assumed that the enzyme concentration
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Holbrook 2003b), in other words gradient remains constant with time, i.e. ∂t ∇e = 0.
|∇ • (D · ∇s)| |κ/µ · RT · ∇ • (s · ∇s)| . 2.1.6 Compartmental modelling. When solutes are transported from
(10)
one compartment to another the difference in compartment volume
This allows phloem translocation to be modelled with the following
needs to be accounted for. A possible way to do this is to have volumes
approximation:
appear explicitly in the relevant equations, with solutes measured in
ṡ − κ/µ · RT · ∇ • (s · ∇s) = vl (z) − vu (z). amounts. Terms in the differential equations would then have units
(11) of amounts per unit time. Therefore, all s in Equations (12)–(14) are
Thompson (2006) called this ‘solute-driven advection’. actually defined as
The next section describes the equations governing catalytic conver-
S 1 No. of moles of S
sions and cross-membrane transport, in other words, finding the rates to s= · =
V S0.5
(15) Volume × Half-saturation constant
populate v in terms of metabolite concentrations and parameters.
and appear as such in the model. Solute flow and reactions are defined
2.1.5 Reactions and cross-membrane transport. To model biochem- on certain compartments and sometimes also span a number of
ical reactions, we use a generic enzyme-kinetic rate equation, i.e. the compartments.
generalized reversible Hill equation, which for a bi-substrate bi-prod-
uct reaction reads (Hanekom et al. 2006; Rohwer et al. 2006, 2007) 2.1.7 Justification for phenomenological equations. Modelling
sometimes requires a choice to be made between using equations of
Å ã (h−1)
v Γ (1 + γα2 mh ) i=1,2 si (si + pi ) state that are mechanistically complete or just behaviourally com-
= 1− · ,
(12)
Vf Keq m·r plete, the latter being a necessary condition for the former. In gen-
eral, the definitive equation for modelling fluid flow would be the
Navier–Stokes equations (Beek et al. 1999). There are many difficul-
with m = [1 + α0 mh , 1 + α1 mh , 1 + α2 mh ] ties in solving these equations and what is gained from the solution is
not necessarily of any practical use. For example, the Navier–Stokes
h h h T equations can account for turbulent flow, but one would not expect
and r = [1, (s1 + p1 ) + (s2 + p2 ) , ((s1 + p1 )(s2 + p2 )) ] , to see turbulence in plant phloem. In other words, Reynolds num-
bers would be very low and flow would be laminar. Similarly, phloem
sap is not really compressible. This would justify the use of simpli-
where s, p and m are, respectively, substrate, product and modifier
fied forms of Navier–Stokes as in Henton et al. (2002). Similarly,
concentrations scaled by their specific half-saturation constants. Vf is
Equation (11) is used as a simplified version of the conservation
the limiting forward reaction rate, Γ is the mass-action ratio and Keq
equation for modelling phloem (Thompson 2006).
is the equilibrium constant for the reaction (the disequilibrium ratio
Γ/Keq determines the direction of the reaction); h is the Hill coeffi-
A similar reasoning applies to enzyme-kinetic rate equations.
Mechanistic rate equations often require a greater number of param-
cient (Hofmeyr and Cornish-Bowden 1997; Hanekom 2006); α and
γ are the scaling coefficients, either activating or inhibiting the enzyme eters than phenomenological ones; moreover, these parameters have
not necessarily been measured. The generalized reversible Hill equa-
through, respectively, a K-effect or V-effect (Cornish-Bowden 2012).
tion introduced above explicitly incorporates a thermodynamic term
The dot product, m · r, is used for convenience.
and thus implicitly satisfies the Haldane relationship, has all kinetic
If α = γ = h = 1 and m = 0 (i.e. there are no allosteric effects
terms separated from thermodynamic terms, and possible allos-
and the enzyme does not show cooperativity), then Equation (12)
teric terms are separated from mass-action terms. Finally, all kinetic
simplifies to
parameters are phenomenological and thus have a direct operational
Å ã
v Γ s1 s2 definition in terms of their experimental interpretation (Hofmeyr and
= 1− · .
(13) Vf Keq (1 + s1 + p1 )(1 + s2 + p2 ) Cornish-Bowden 1997; Rohwer et al. 2006).
For these reasons we have taken a purely phenomenological
The equation for uni-uni reactions is still simpler: approach in the formulation presented above, i.e. the Darcy equation6 • Uys et al.
for modelling flow in porous media Bear (1972) to model phloem flow, is not meant to refer to protons in a mechanistic way, but is included
and the generalized reversible Hill equation to calculate reaction rates. to generically couple the uptake of Ssk into the vacuole to a secondary
concentration gradient. In our opinion this level of abstraction is justi-
2.2 Computational methods fied in the core model presented here, the purpose of which is primar-
Computational analyses were performed on a laptop computer using ily to demonstrate the feasibility of the modelling approach.
the Python programming language with the open-source numpy Metabolic activity can regenerate the moieties (e.g. reaction 11 in
(Oliphant 2006), scipy (Virtanen et al. 2020) and FiPy (Guyer et al. Fig. 2). A feedback mechanism is also present, reminiscent of the allos-
2009, 2017) libraries. FiPy is a PDE solver written in Python using teric inhibition of phosphofructokinase by ATP (Evans et al. 1981). The
finite volume methods, and provides an object-oriented interface to import of Ssk into the vacule is via a proton antiporter (Reinders et al.
solving coupled sets of PDEs. Suppose a problem can be cast in the 2006). The sink component of the reaction network in the model has
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
form of a general conservation equation, a number of distinguishing features. The reactions occur in more than
one compartment (Welbaum and Meinzer 1990; Beevers 1991; Rae
∂(ρϕ) n et al. 2005a; Kruger et al. 2007), in contrast to previous models where
= ∇ · ( uϕ) + [∇ · (Di ∇)] ϕ + Sϕ ,
(16)
∂t only subcellular compartment was considered (Rohwer and Botha
2001; Uys et al. 2007). It is possible for some metabolite intermedi-
where ρ is some coefficient, ϕ is the variable being solved, u is a vector
ates to enter futile cycles (Komor 1994; Zhu et al. 1997), while other
coefficient (e.g. velocity), Di is a generic coefficient of diffusion (e.g.
metabolites may form part of moiety-conserved cycles (e.g. ATP/ADP)
heat or solute diffusion) and Sϕ is a source term (i.e. a source of ϕ ),
(Dancer et al. 1990; Cloutier et al. 2009). Assimilate can be partitioned
then there is a natural way to program the equation in Python. It is
between respiration or storage (Whittaker and Botha 1997; Bindon
clear that Equation (3) is amenable to this formalism.
and Botha 2002), where accumulation is due to energy-driven anti-
Simulation data were visualized with the matplotlib plotting library
porter proton gradients (Thom and Komor 1985; Reinders et al. 2006).
(Hunter 2007).
Stored assimilate can also be remobilized from the vacuole (Moore
1995). Energy is captured from metabolite breakdown and consumed
3. R E S U LTS by synthesis and transport steps, reactions in the respiration branch are
3.1 Example of building a model subject to negative allosteric feedback (i.e. synonymous with phospho-
Previous models of phloem flow have usually approximated the sieve fructokinase [PFK] and pyruvate kinase [PK]) and these reactions are
tube as a cylinder with loading of solute at the one end and unloading at positively cooperative.
the other (Henton et al. 2002; Thompson and Holbrook 2003a). A few The inclusion of moiety-conserved cycles such as ATP/ADP in a
models have also considered branched vascular bundles (Lacointe and model can sometimes lead to the introduction of constraints on the
Minchin 2008). Some others have included general sink type demand, fluxes because the moiety must be balanced at steady state. Such con-
such as ‘respiration’ or ‘starch’ (Quereix et al. 2001). The aim of this straints may be unintentional or artefactual, and one way of counteract-
paper is to combine phloem modelling including loading and unload- ing a possible side effect is to introduce an additional ATP-consuming
ing along the entire length of the sieve tube with compartmentation or -producing reaction. This was the reason for including reaction 12.
and detailed reaction kinetic modelling.
3.1.2 Partial differential equations. Table 2 summarizes the processes
3.1.1 Pathway. The model is constructed to emulate a single sug-
that were accounted for in each compartment of the combined model.
arcane stalk. A segment of the whole pathway, a single node/inter-
Advection–diffusion is assumed only to occur in the phloem and is dom-
node pair, is shown in Fig. 2. The primary storage organ is the stalk
inated by the advection component, as derived in Equation (11) above.
(in particular the vacuoles of the parenchymal cells) and sources of
Species in the symplast can diffuse, be transported or react; source and
assimilate are distributed along the stalk. There are five nodes and
vacuole by contrast are assumed to be ‘well-stirred’. Thus, depending
internodes, which represent immature to mature tissue. The primary
on which compartment is being modelled, the appropriate terms from
storage organelle is the vacuole, with possible assimilate remobiliza-
Equation (3) for the various processes are included or ignored.
tion. A futile cycle known to exist in sugarcane (Komor 1994) is mod-
Keeping the above in mind, the equations governing the system are
elled explicitly; sucrose can be broken down into fructose and glucose.
divided into four groups. The first group describes those species that
The hexoses can then enter the hexose phosphate pool and sucrose
are only involved in reactions; in other words, species found in the leaf
can be synthesized again. However, the cycle can either take place in
and vacuole:
the symplast alone (i.e. the cytosolic space inside the plasma mem-
brane formed by linking of the cytoplasm of successive cells through ∂t Sso = v1 − v2 ,
plasmodesmata), or across the symplast and vacuole. The apoplast
(i.e. the extracellular space outside the plasma membrane formed by
cell walls of adjacent cells) was not included as a separate compart- ∂t Svc = v8 − v6 ,
ment in this first version of the model. The model also includes two
moiety-conserved cycles (Msk/Nsk, representative of ATP/ADP; and ∂t Pvc = v6 − v9 .
Hsk/Hvc, referring to the movement of protons across the tonoplast, i.e.
the membrane surrounding the vacuole). Importantly, the species Hsk With the following shorthand,Coupling kinetic models and advection–diffusion equations 1 • 7
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Figure 2. A slice through the model pathway showing a node/internode pair. Metabolites Xso and Zsk are clamped; in other words,
their concentrations are treated as constant. All other metabolite concentrations are free to vary. Msk and Nsk behave like a moiety-
conserved pair, rather like ATP/ADP. Reaction 8 is a proton (Hsk) antiporter. Futile cycling refers to the energy-consuming
breakdown and synthesis of a metabolite, in this case Ssk. Reactions 10 and 11 are reminiscent of phosphofructokinase and
pyruvate kinase, both of these are allosterically inhibited by ATP (Msk in the case of the model).
∂ ∂2 ∆Hsk = v8 − v7 .
∆X = X + DX 2 X,
∂t ∂z
Note that the stoichiometry for reaction 10 is 1P + 2M → 1Y + 2N
the PDEs governing species behaviour in the symplast are: and for reaction 11 it is 1Y + 4N → 1Z + 4M . Reaction 10 is repre-
sentative of upper glycolysis, where two ATP are consumed, and reac-
∆Ssk = v3 + v4 − (v5 + v8 ), tion 11 of lower glycolysis where four ATP are produced. All other
reactions have unitary stoichiometry. The specific case of assimilate
∆Psk = v5 + v9 − (v4 + v10 ), translocation in the phloem is modelled by
∂t Sph − κ/µ · RT · ∂z (Sph (∂z Sph )) = v2 − v3 .
(17)
∆Ysk = v10 − v11 ,
The stoichiometric model in Fig. 2 has two conservation relations describ-
ing the total amounts of Msk + Nsk and Hvc + Hsk . These conservation
∆Msk = 4 · v11 − (v12 + 2 · v10 + v4 + v7 ), relations were maintained with the following algebraic expressions:8 • Uys et al.
Table 2. Processes allowed for each compartment Molecules in the symplast can move up, down, into the phloem or into a
vacuole. A vacuole corresponds to one finite volume; consequently there
Advection Diffusion Facilitated Reaction are 50 vacuolar compartments in the symplast. Molecules in a vacuole
transport
cannot move to another vacuole, they can only move to the symplast.
Source ● ● The choice of N = 50 was determined by the minimum number of
Phloem ● ● ● finite volumes required to have five node and internode pairs such that the
Symplast ● ● ● length of an internode was nine times longer than a node, as approxima-
Vacuole ● ● tion of real internode lengths (van Dillewijn 1952).
Because this was a core model, any units assigned to param-
eters and variables would be arbitrary to some extent. However,
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Nsk = C1 − Msk , it should be emphasized that variables were defined in amounts,
(18) so concentrations—and by implication also half-saturation con-
stants—were in amount per volume. Rates take their units from
Hvc = C2 − Hsk . the maximal activities and these were in amount per unit time,
(19)
not concentration per unit time. Each of the finite volumes was
The vi appearing on the right-hand side of the PDEs are themselves
assumed to have identical size across the different compart-
functions and are determined by the rate equations (Equation (12)).
ments, obviating the need for correction to kinetics for movement
between volumes of different size. For the compartments allowing
3.2 Discretization and parameter selection advection and/or diffusion (phloem and symplast), the simulated
3.2.1 Geometry and domain The first step in building the model
concentration of a species assumed the same value throughout a
was to describe the geometry, which needed to be divided into finite
particular finite volume; this is the consequence of the discretiza-
volumes or be ‘meshed’. Compartments were then defined on these
tion of the continuous PDEs.
elements. A compartment can span a number of elements and com-
The ‘ground state’ for the model had all parameters set to a value
partments of the same type can be separated by a number of elements.
of 1—so-called ‘vanilla’ kinetics. Any changes to the ground state
A pathway or reaction network was defined across the compartments.
were only introduced to account for a certain behaviour. Furthermore,
Species concentrations and reaction rates are variables that in turn were
parameters were chosen to exaggerate the desired behaviour, as dis-
defined on the network. Note that the same chemical species found in
cussed below.
two different compartments was modelled as two separate variables.
Therefore, each variable was only defined for its specific compartment
and could not assume a value anywhere outside of it (the domain of 3.2.2 Thermodynamic parameters. Equilibrium constants were cho-
the variable). In the model, reaction rates were calculated from species sen with the following points in mind:
concentrations and it follows that they cannot have values at positions
1. the preferred direction in which a reaction was to occur,
where a species variable does not exist. The behaviour of the variables
2. adherence to the Haldane relationship,
is governed by a set of PDEs. Once the model formulation had pro-
3. adherence to microscopic reversibility and detailed
ceeded this far, a time-course simulation could be performed given suf-
balance, and
ficient parameter values, initial conditions and boundary conditions.
4. dependencies between equilibrium constants for coupled
The model behaviour was then analysed with appropriate software and
reactions.
programs.
Given the above hierarchy of modelling steps, we started with a Point 1 was addressed by not assuming any preference for substrates or
straight-line segment to approximate the domain by a 1D mesh with products, except in the case where ‘energy equivalents’ were produced
arbitrary length, L = 1.0, and divided into 50 finite volumes. The com- or consumed. The use of generic reversible Hill equations implicitly
partments, i.e. source (leaf), phloem, symplast and vacuole, were then satisfied Point 2.
defined on these finite volumes (Fig. 3). Each finite volume element has The concepts of detailed balance and microscopic reversibility
at least three compartments, i.e. the phloem, symplast and vacuole. Some require that the product of the equilibrium constants around a closed
finite volume elements have an additional fourth compartment, i.e. the cycle be equal to 1. Points 3 and 4 are closely related. Consider Fig. 2,
source. The phloem and symplast span all 50 elements, with leaves act- starting at the futile cycle in the model. Reaction 5, S → P , is the
ing as source tissue feeding assimilate into the phloem at the nodes, and opposite of the half reaction, P → S , of enzyme 4. By definition,
storage parenchyma in the internodes (symplast) acting as sink tissues KSP = KPS −1; therefore, KSP KPS = 1. It is possible for reaction 4 to
extracting assimilate from the phloem (see Fig. 3). There are five sequen- have an equilibrium constant larger than KPS because it is coupled to
tial node and internode pairs, numbered from 1 to 5, where internode 1 the half reaction, M → N , which does have a large equilibrium con-
is considered immature tissue and internode 5 is mature tissue. There are stant so that K4 = KPS KMN can be large. Note, however, that the
10 elements to a node/internode pair; the first element of each group of interconversion of M and N (in both directions) is involved in other
10 is used for the node, the remaining nine for the internode. The only reactions as well, which places a constraint on the choice of equilib-
path out of a leaf element is into the phloem. There are four possible paths rium constants for those other reactions as the equilibrium constant
for a molecule in the phloem, up, down, into the leaf or into the symplast. for this specific half reaction may not change.Coupling kinetic models and advection–diffusion equations 1 • 9
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Figure 3. Geometry and domain discretization of the model. The sugarcane stalk was approximated in 1D and compartments
defined as indicated. Finite volumes of identical size are defined on each compartment; for the phloem and symplast these are
shown by dotted lines because solutes can be transported along these continuous compartments by advection and/or diffusion as
specified in Table 2.
The easiest way to satisfy all these criteria was to spec- of the phloem loading step. They were chosen to be the highest activi-
ify the equilibrium constants for the half reactions as follows: ties, because there are only five points along the stalk that produce
KSP = KPS = 1, KPY = 0.1, KYZ = 104 , and KMN = 10 (by implica- assimilate through phloem loading, but 50 that consume it through
tion KNM = 0.1). The high value of KYZ allows reaction 11 to provide phloem unloading. The chosen value was Vf 1,2 = 10 .
the driving force for the regeneration of the moiety N → M . The The second group of activities chosen to remain constant were
equilibrium constant for the half reaction of transport of a single spe- those of the two reactions involved in the futile cycle, the active proton
cies across compartmental boundaries (i.e. not considering any energy transport across the tonoplast and the remobilization step from the
coupling) is 1 by definition. With these choices, equilibrium constants vacuole—reactions 4, 5, 7 and 9. These were kept in the vanilla state,
for all reactions are equal to 1, except for reactions 4, 7, 10 and 12, i.e. = 1.
which have a value of 10 (they are ‘energy-driven’ reactions). The maximal activities that were modelled to change with increas-
ing z are shown in Fig. 4. Recall that z is defined on the interval
3.2.3 Kinetic parameters All substrate and product half-saturation z ∈ [0, L] and that L = 1 in this geometry. The activities that increased
constants were set to 1. For reactions 10 and 11, α = γ = 0.8 and the with increasing z are the phloem unloading steps and the vacuolar
modifier half-saturation constant was 5. The Hill coefficient in both accumulation steps.
reactions was set to 4, because both PFK and PK are tetramers. In real- The rough guiding principle in selecting parameters was to observe
ity the Hill coefficients are frequently lower, as the number of subunits accumulation of Svc along the stalk and with time. The ground state
represents an upper bound on the numerical value the Hill coefficient parameter set (all values = 1) unsurprisingly did not show this, and
can take. However, as mentioned, the idea behind this model was to we tried to induce this behaviour with as few parameter changes as
provide a simplified, if slightly exaggerated, representation with which possible.
the behaviour of the system could be simulated in a reasonable amount
of CPU time. 3.2.4 Flow coefficients. Diffusion coefficients were set at the same
Amongst the maximal activities in the model that remained con- value of 10−5 for all species in the symplast. The flow coefficient was
stant with increasing z are those of assimilate synthesis in the leaf and assigned a value of10 • Uys et al.
(20) slower than the reaction rates or rate of facilitated transport, then spe-
κ/µ · RT = 1.0 × 10−4 ,
cies form a heap round the nodes.
such that the diffusion coefficient in the symplast was at least one order ‘Sucrose’: The highest concentration throughout the simulation
of magnitude smaller than the flow coefficient in the phloem. Since dif- was that of the ‘sucrose’ analogue Svc, as can be seen on the left-hand
fusion and translocation occur in opposite directions with reference side of Fig. 5. Sso concentrations were initially evenly distributed along
to a concentration gradient, the convention is to choose negative flow the stalk, and end the simulation at a shallow gradient monotonically
parameters, so that fluxes are positive. In other words, if the concentra- decreasing with z . Similarly Sph in general decreased down the stalk,
tion gradient has negative slope, then the direction of flow is positive. except for the local peaks around the nodes. The Ssk concentration
profile developed from an initial linearly increasing distribution to
3.2.5 Boundary and initial conditions. Boundary conditions were one that became constant with z—once gain with the exception of
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
applied at z = {0, L} such that there is a zero gradient at the bounda- the node peaks. Towards the end of the simulation, the concentrations
ries (e.g. for state variable S , ∂z S(0) = 0 and ∂z S(L) = 0 ), ensuring of S decreased from leaf, to phloem, to symplast and then increased
zero flux over the boundaries (the default boundary conditions in again in the vacuole. In other words, assimilate flow was down a con-
the FiPy software). Such boundary constraints are a special case of centration gradient. Svc increased along the stalk, with concentrations
Neumann boundary conditions; see the Discussion for a distinction at immature end climbing faster than those at the mature end.
between Dirichlet (fixed value) and Neumann (fixed gradient) bound- Moieties: The simulation profiles of moiety pairs, Msk and Nsk as well
ary conditions. as Hsk and Hvc, form mirror images across z and t . This is due to the
The initial values of each species variable, given in amounts, are moiety conservation constraints, i.e. the pairs always have to sum up to
summarized in Table 3. The amounts of the moiety-conserved species a constant (Equation 21).
add up to a constant sum as follows: Reaction rates: As with the concentrations, most of the reaction
rates settled into stable profiles (at approximately t > 150 ). The rate
Msk + Nsk = C1 = 9 + 1 = 10, of synthesis (v1) in the leaf increased with z and t ; this was due
to decreasing Sph. Similarly phloem loading (v2 ) increased because
Hsk + Hvc = C2 = 1 + 99 = 100. of increasing Sso and decreasing Sph. Phloem unloading (v3) fell to
(21) a level much lower than the loading rate, because only five source
elements have to supply a total of 50 sink elements (recall that an
3.3 Simulation results element with a node defined on it overlaps a part of the symplast as
This section summarizes the development of concentration and reac- there is parenchymal tissue at the nodes as well as in the internodal
tion rate profiles during the time simulation. The results are presented regions, see Fig. 3).
in Figs 5 (concentration profiles) and 6 (reaction rates). The rates of the cooperative, allosteric reactions v10 and v11
Sawtooth profiles: The distributions of Sph, Ssk, Svc, Psk and Pvc along stabilized relatively quickly, reflecting the allosteric inhibition by
the stalk all exhibited ‘sawtooth’ profiles to some extent. These peaks Msk—homoeostasis was almost immediate. Since v11 is respon-
correspond to the node positions; this is where all the assimilate is sible for regenerating the energy equivalents, homoeostasis will
being loaded and phloem unloading rates are the highest. In the model, impact the vacuolar uptake step. Furthermore, the moiety regen-
species are advected (in the phloem) or diffuse (in the symplast) away eration and consuming reactions were very close to a steady state,
from the nodes. If the rate of translocation, advection or diffusion is i.e. 4v11 − (2v4 + v7 + v10 )≈0 for the entire length of the stalk at
t = 600 time steps.
The rate v6 had a very pronounced negative parabolic profile that
gradually disappeared within the first 150 time points. This is dis-
cussed in detail in the Discussion section and Fig. 8.
Futile cycling: The reactions v4 and v5 form a futile cycle in the
symplast. The calculated rate of v5 was higher than v4 ; therefore,
more Ssk was being broken down than regenerated. Moreover, S is
also broken down in the vacuole by v6 , adding to the overall degree
of futile cycling as illustrated in Fig. 6. Note that in spite of this
futile cycling ratio of > 1 it was still possible for Svc to increase
over time, because S is additionally also taken up into the symplast
through v3.
Figure 4. The change in maximal activities across stalk length
3.4 The ‘sock experiment’
Increased photosynthetic rates in response to shading have been
for reactions 3, 6, 8, 10 and 11. The functions to the left and
right of the plot are the straight-line equations describing the experimentally demonstrated in sugarcane (McCormick et al.
maximal activity profile as a function of z . 2006, 2008); shading of the leaves was argued to correspond toCoupling kinetic models and advection–diffusion equations 1 • 11
Table 3. Initial amounts of each species in each compartment
X S P M N H Y Z
Source 100 70 0 0 0 0 0 0
Phloem 0 20 0 0 0 0 0 0
Symplast 0 14z + 1 2 9 1 1 2 1
Vacuole 0 140z + 10 10 0 0 99 0 0
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
Figure 5. Concentrations for 615 time steps, from a simulation of the model in Fig. 2. Sso is only defined at a node and therefore
appears as ‘bars’. Those concentrations with a ‘sawtooth’ profile each have a small peak corresponding to a node position. Note
that all the S species showed high concentrations.12 • Uys et al.
an increase in sink demand. All but one leaf of a sugarcane stalk
10 z = 0, 0.2, 0.4, 0.6, 0.8 10 z = 0.2
was covered with shade cloth and the effect on various sugars Vf 1 (z) = → Vf 1 (z) =
0 otherwise 0 otherwise
and rates measured. Sucrose concentrations decreased in young
internodal tissue, while the carboxylation efficiency and rate of (22)
electron transport increased in the exposed leaf (McCormick In other words, all but the second leaf had the maximal velocity for
et al. 2006). Expression of a number of genes associated with reaction 1 set to zero (Fig. 7). The following effects can be noted after
photosynthesis, mitochondrial metabolism and sugar transport the ‘shading’ event, compared to the reference model:
were also upregulated (McCormick et al. 2008).
We investigated whether this phenomenon could be captured with • v1 (sucrose synthesis in the leaf) and v2 (phloem loading)
our simplified core model. To model the effect of shading, the follow- in the second (i.e. non-shaded) leaf increased above levels of
the reference model. They remained higher for the rest of
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
ing event was triggered at t = 102 during a time-dependent simulation
of the reference model. the simulation.
Figure 6. Reaction rates for 615 time steps, from a simulation of the model. Note the parabolic profile of reaction 6, this is
discussed in Fig. 8 and the text.Coupling kinetic models and advection–diffusion equations 1 • 13
• P accumulated more slowly than in the reference model. specific solute using known experimental measurements. The usual
• Metabolite concentration profiles decreased more rapidly way of increasing the accuracy of the van’t Hoff equation is by using
with z compared to the reference model. a virial-like expansion for non-ideal cases, but the virial coefficients
• Assimilate flowed up and down the stalk away from the only need to be measured by experiment (Atkins and de Paula 2002).
remaining source. A phenomenological equation has been proposed by Michel (1972)
and is used by Thompson and Holbrook (2003a), Minchin and co-
workers (Lacointe and Minchin 2008; Hall and Minchin 2013) and
4. D I S C U S S I O N Seleznyova and Hanan (2018). Lacointe and Minchin (2008) calcu-
In this paper we have presented a general ADR framework for model- lated that the error between a first and higher order approach is in the
ling phloem flow in plants coupled to biochemical reactions and trans- order of 8 %. Such an error is insignificant for the simplified model
Downloaded from https://academic.oup.com/insilicoplants/article/3/1/diab013/6206762 by guest on 06 November 2021
port across compartment boundaries. The framework was applied to with arbitrary parameter values presented here, and we therefore did
a simplified model of sucrose accumulation in sugarcane. While the not adopt the more complicated approaches.
modelling of both phloem and xylem flows has a long history ( Jensen
et al. 2016), and recently has been extended to a whole-plant frame- 4.2 Model
work (termed CPlantBox, Zhou et al. 2020) for modelling water and The model formulation in Fig. 3 is of course a simplification of what
carbohydrate flows, to our knowledge this is the first report combin- happens in a sugarcane plant, and the formulation could be criticized
ing explicit simulation of enzyme-catalysed reactions within the ADR on this ground. In what follows, we provide a justification for our sim-
framework. plifying choices.
In the following we first discuss the mathematical framework, fol- First, the intricate network of reactions in the leaf is abbreviated
lowed by a critical evaluation of the model description and further dis- to one reaction in the model. This assumption is partly based on the
cussion of some simulation results. finding that photosynthetic rates respond to changes in demand rather
than the other way around (Wu and Birch 2007; McCormick et al.
4.1 Framework 2008).
Equation (3) is a general framework that allows the modelling of Next, the model does not account for growth or volume changes
metabolites that are involved in chemical reactions, advected or dif- with time. Large time scales therefore do not give an accurate picture
fused. It is a broad framework, which requires that an expression for of what happens in the plant. To model growth a method such as ‘adap-
fluid velocity be provided, compartmentation be defined, rate equa- tive meshing’ would be required to dynamically adapt the domain to
tions and stoichiometry be specified and a suitable geometry with the change in geometry (Plewa et al. 2005). Partial differential equa-
boundary conditions be provided. This was successfully applied to a tions that account for volume changes and partitioning of carbohy-
pathway of biochemical reactions. By allowing such a pathway to vary drates to structural components would be needed as well; however,
spatially, more complicated behaviour can be modelled compared to a this was beyond the scope of out initial ‘proof-of-concept’ of the ADR
well-mixed case. Furthermore, the model provides a good first approxi- framework.
mation of the modelling of sugarcane geometry. Third, there is an achievable method of modelling growth with
There are, of course, a number of limitations in this approach. First, the current formulation, namely by altering the Neumann bound-
the use of Darcy’s law to model fluid flow could be interpreted as an ary conditions. Suppose y = f (x, ∂x) is some function of interest on
oversimplification. Darcy’s law is a phenomenological equation that the interval [a, b], then the Dirichlet boundary condition would be
describes fluid flow in porous media. It requires flow parameters to be y(a) = c1 and y(b) = c2, while the Neumann boundary condition is
measured. The use of Darcy’s law is justified where macroscopic flow slightly more complicated, ∂x y(a) = c3 and ∂x y(b) = c4 . The model-
is modelled, i.e. it is entirely inappropriate for modelling fluid flow in ling approach thus far has been to set the gradient of all state variables
a single sieve element, but adequate for modelling flow through many to zero at the boundaries, ensuring zero flux across the boundaries.
sieve elements strung together. A statistical averaging effect is at work Although changing the fixed gradient to a non-zero value would not
over sufficiently large scales, and phloem flow and translocation over a explicitly model growth, the supply or demand for reactants could
whole plant qualifies as such. Despite its limitations, the use of Darcy’s then take place across the boundaries. Furthermore, this exchange can
law has advantages: for example, it is possible to simplify Equation change with growth. This is left for future work.
(3) by taking into account the mechanism of phloem translocation Fourth, the model discretization in terms of a 1D mesh is of course
(Thompson 2006). This simplification is easier to solve, because, first, a simplification. For the sugarcane stalk, which is unbranched, this
the convective term is removed, and second, no pressure gradient has is an appropriate approximation; the same argument could be made
to be calculated. The simplification also describes the macroscopic (or for constructing similar models for other grasses. The approximation
phenomenological) aspects of phloem translocation, i.e. that flow is is, however, insufficient for other plants that show branches in their
due to a pressure gradient induced by a solute. stems. In this case, a 2D or 3D mesh would have to be defined that
The use of the van’t Hoff equation is a further simplifying assump- appropriately approximates the physical structure to be modelled.
tion. This equation is usually a first approximation for the calculation While this is tractable in principle, and the FiPy software has built
of osmotic pressure, since it only applies to dilute solutions. However, in support for various 2D and 3D meshes as well as custom meshes
the only viable alternative is to use an interpolative function for a generated by Gmsh (Geuzaine and Remacle 2009), a custom 3D finiteYou can also read
