Journal of Theoretical Biology 432 (2017) 109–131

Contents lists available at ScienceDirect

Journal of Theoretical Biology

journal homepage: www.elsevier.com/locate/jtbi

Mathematical modelling and analysis of the brassinosteroid and

gibberellin signalling pathways and their interactions

Henry R. Allen

a , Mariya Ptashnyk

a , b , ∗

a Department of Mathematics, Fulton Building, University of Dundee, Dundee, DD1 4HN, United Kingdom

b Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

a r t i c l e i n f o

Article history:

Received 19 March 2017

Revised 28 July 2017

Accepted 13 August 2017

Available online 14 August 2017

2010 MSC:

34Cxx

35Q92

65Nxx

92Bxx

92Cxx

Keywords:

Plant modelling

Hormone crosstalk signalling

Homeostasis in plants

Stability analysis and Hopf bifurcation

PDE-ODE systems

a b s t r a c t

The plant hormones brassinosteroid (BR) and gibberellin (GA) have important roles in a wide range of

processes involved in plant growth and development. In this paper we derive and analyse new mathe-

matical models for the BR signalling pathway and for the crosstalk between the BR and GA signalling

pathways. To analyse the effects of spatial heterogeneity of the signalling processes, along with spatially-

homogeneous ODE models we derive coupled PDE-ODE systems modelling the temporal and spatial dy-

namics of molecules involved in the signalling pathways. The values of the parameters in the model for

the BR signalling pathway are determined using experimental data on the gene expression of BR biosyn-

thetic enzymes. The stability of steady state solutions of our mathematical model, shown for a wide

range of parameters, can be related to the BR homeostasis which is essential for proper function of plant

cells. Solutions of the mathematical model for the BR signalling pathway can exhibit oscillatory behaviour

only for relatively large values of parameters associated with transcription factor brassinazole-resistant1’s

(BZR) phosphorylation state, suggesting that this process may be important in governing the stability of

signalling processes. Comparison between ODE and PDE-ODE models demonstrates distinct spatial distri-

bution in the level of BR in the cell cytoplasm, however the spatial heterogeneity has significant effect

on the dynamics of the averaged solutions only in the case when we have oscillations in solutions for

at least one of the models, i.e. for possibly biologically not relevant parameter values. Our results for the

crosstalk model suggest that the interaction between transcription factors BZR and DELLA exerts more

influence on the dynamics of the signalling pathways than BZR-mediated biosynthesis of GA, suggesting

that the interaction between transcription factors may constitute the principal mechanism of the crosstalk

between the BR and GA signalling pathways. In general, perturbations in the GA signalling pathway have

larger effects on the dynamics of components of the BR signalling pathway than perturbations in the BR

signalling pathway on the dynamics of the GA pathway. The perturbation in the crosstalk mechanism

also has a larger effect on the dynamics of the BR pathway than of the GA pathway. Large changes in the

dynamics of the GA signalling processes can be observed only in the case where there are disturbances

in both the BR signalling pathway and the crosstalk mechanism. Those results highlight the robustness

of the GA signalling pathway.

© 2017 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

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. Introduction

The sessile nature of plant life highlights the importance of ef-

cient regulatory mechanisms allowing plants to respond to envi-

onmental stimuli and to adapt to changing environmental condi-

ions. Plants have developed a set of highly integrated signalling

athways. Plant hormones, e.g. auxin, gibberellin, cytokinin, brassi-

∗ Corresponding author.

E-mail addresses: m.ptashnyk@dundee.ac.uk , mptashnyk@maths.dundee.ac.uk

M. Ptashnyk).

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ttp://dx.doi.org/10.1016/j.jtbi.2017.08.013

022-5193/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article u

osteroids, ethylene, are key signalling molecules and their activi-

ies depend on the cellular context and interactions between them.

The family of steroidal plant hormones brassinosteroids (BRs)

s responsible for the regulation and control of a wide range of es-

ential processes including responses to stresses ( Bajguz and Hayat,

009; Gruszka, 2013 ), photomorphogenesis ( Belkhadir and Jaillais,

015; Zhu et al., 2013 ), root growth ( Müssig et al., 2003 ), and stom-

tal development ( Kim et al., 2012 ). Over the last 47 years, the

ffects of brassinosteroids on plant cells and plants as a whole,

s well as their signalling pathways have been studied in detail

Clouse, 2015 ). In particular the signalling pathway of brassino-

ide (BL), the most biologically active of the discovered BRs, has

nder the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

110 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

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been examined in great detail, and is now one of the most un-

derstood pathways in plant biology ( Belkhadir and Jaillais, 2015;

Clouse, 2011; Kim and Wang, 2010; Li and Deng, 2005; She et al.,

2011; Wang et al., 2014c; 2001; Yang et al., 2011; Zhu et al., 2013 ).

BR signalling functions by controlling the expression of various

genes regulating developmental processes, of which 10 0 0s have

been identified ( Sun et al., 2010 ), including several genes which

regulate the production of proteins that act as enzymes during

BR biosynthesis ( Tanaka et al., 2005 ). To ensure controlled growth,

homeostasis of BRs in plant tissue is carefully maintained by neg-

ative feedback in the BR signalling pathway ( Tanaka et al., 2005 ).

Absence of BRs and/or perturbed BR signalling have also been

linked to many growth defects, including dwarfism and male steril-

ity ( Clouse, 1996; Clouse and Sasse, 1998 ).

Gibberellins (GAs) are another family of plant hormones in-

volved in many developmental processes in plants, including seed

germination, stem elongation, leaf expansion, trichrome develop-

ment, pollen maturation and the induction of flowering ( Achard

et al., 2008; Davière and Achard, 2013 ). There are over 130 cate-

gorized gibberellins, a few of which are bioactive molecules, the

most common being GA 1 , GA 3 and GA 4 ( Yamaguchi, 2008 ).

Along with detailed information on individual plant hormone

signal transduction pathways, their target genes, and their effects,

it is now also known that interactions between various molecules

involved in different signalling pathways have an effect on physi-

ological phenomena in plants ( Bai et al., 2012; Belkhadir and Jail-

lais, 2015 ). For example both auxin and brassinosteroids play a role

in the patterning of vascular shoot bundles ( Ibañes et al., 0 0 0 0 ),

as well as exhibiting some cross-regulation of biosynthesis via the

BRX gene ( Sankar et al., 2011 ). The GA signalling pathway exhibits

a high level of interaction with the BR signalling pathway regu-

lating growth ( Clouse and Sasse, 1998; Úbeda Tomás et al., 2009 )

and responses to stresses ( Achard et al., 2008; Bajguz and Hayat,

2009 ) among others. The interactions between different signalling

pathways are called crosstalks, and the investigation of their mech-

anisms is key to better understand plant hormonal responses to

external stimuli ( Albrecht et al., 2012; Belkhadir and Jaillais, 2015;

Gruszka, 2013; Yang et al., 2011 ). Despite the need for better un-

derstanding of the mechanisms of crosstalk between hormone sig-

nalling pathways, it is hard to obtain experimentally quantitative

data on the dynamics of all molecules involved in the signalling

pathways and interactions between them. For example, it is very

difficult to measure the dynamics of hormones such as BR in real

time, in part due to their occurring naturally at extremely low

levels. Therefore development of accurate mathematical models of

hormone activity can help to analyse and better understand in-

teractions between signalling pathways and their impact on plant

growth and development. Hence, the main aim of this paper is

to derive and analyse novel mathematical models for the BR sig-

nalling pathway, and the crosstalk between the BR and GA sig-

nalling pathways.

Along with many modelling results for the gibberellin and

auxin signalling pathways ( Gordon et al., 2009; Liu et al., 2010;

Middleton et al., 2010; 2012; Muraro et al., 2011 ), only a few

models can be found for BR-related signalling processes. A logic

model for the activation states of the components of the BR and

auxin signalling pathways and their interactions was derived in

Sankar et al. (2011) , and provided a qualitative description of the

dynamics of the BR pathway. In a simple model for BR-mediated

root growth, proposed in van Esse et al. (2012) , the growth dynam-

ics is assumed to be dependent on the quantity of receptor-bound

BL, which was considered to be constant. A system of ordinary dif-

ferential equations was considered to describe the dynamics of BR-

regulated transcription factor BES1 (BRI1-EMS SUPRESSOR 1) and

its interactions with R2R3-MYB transcription factor BRAVO, related

to division of plant stem cells ( Frigola et al., 2017; Vilarrasa-Blasi

t al., 2014 ). To our knowledge there are no previous results on

athematical modelling of the crosstalk between BR and GA sig-

alling pathways.

In our mathematical model for the BR signalling pathway, we

onsider the dynamics of BR, free and bound receptors, inhibitors,

nd phosphorylated and dephosphorylated transcription factors,

ot considered in previous models. The spatially homogeneous dy-

amics of the molecules involved in the signalling pathway that

e consider are modelled by a system of six ordinary differential

quations (ODEs). To analyse the effect of spatial heterogeneity of

he signalling processes on the dynamics of BR, we derive a cou-

led model composed of partial differential equations (PDEs) for

R, inhibitor, and phosphorylated transcription factor, ODEs for re-

eptors, defined on the cell membrane, and the ODE for dephos-

horlyated transcription factors, localised in the cell nucleus. Along

ith spatial distribution of the concentration of BR in the cell cyto-

lasm, we observe similar dynamics for solutions of the ODE and

veraged solutions of the PDE-ODE models when those solutions

onverge to a steady state as t → ∞ . However spatial heterogene-

ty has significant effect in the case when at least one of the two

odels has periodic solutions, which is determined for possibly bi-

logically not relevant parameter values.

To model the crosstalk between BR and GA signalling path-

ays we first rigorously derive a reduced model for the GA sig-

alling pathway from the full GA signalling pathway model pro-

osed in Middleton et al. (2012) . Then we couple the model for the

R signalling pathway with the reduced model for GA signalling

athway by considering three different interaction mechanisms be-

ween BR and GA pathways. By analysing the effect of different in-

eraction mechanisms on the dynamics of molecules involved in

he signalling processes, we determine that one of these mecha-

isms has a more significant effect on the dynamics of the sig-

alling pathways than the other mechanisms. Similar to the BR

ignalling pathway model, we also consider the influence of spatial

eterogeneity of the signalling processes on the dynamics of solu-

ions of the BR-GA crosstalk model. Using the mathematical mod-

ls developed here, we analyse how interactions between the BR

nd GA signalling pathways depend on the model parameters and

he strength of interaction mechanisms. We observed that, in gen-

ral, parameter changes in both pathways have a stronger effect on

he components of the BR signalling pathway than on the compo-

ents of the GA signalling pathway. Our results also suggest that

he interaction between transcription factors exerts more influence

n the dynamics of the signalling pathways than BR signalling-

ediated GA biosynthesis. Further, our results suggest that pertur-

ations in the GA signalling pathway have larger effects on the dy-

amics of components of the BR signalling pathway than the per-

urbations in the BR signalling pathway on the dynamics of com-

onents of the GA signalling pathway, apart from in the case when

e have disturbances in both the BR signalling pathway and the

rosstalk mechanism.

The structure of this paper is as follows. In Section 2 a biological

verview of the BR and GA signalling pathways, and their interac-

ions is given. In Section 3 we derive the mathematical model for

he BR signalling pathway and estimate the values for the model

arameters using experimental data from Tanaka et al. (2005) . In

ection 4 we perform qualitative analysis of the model for the BR

ignalling pathway, examining how the behaviour of solutions of

he mathematical model depends on the values of the model pa-

ameters. We also define the set of parameters for which the sys-

em of ODEs has stable stationary solutions and the set of pa-

ameters for which it undergoes Hopf bifurcation. In Section 5 we

xtend our model for the BR signalling pathway to examine the

ffects of spatial heterogeneity in the signalling processes. In

ection 6 we consider the reduction of the mathematical model

or the GA signalling pathway, proposed in Middleton et al. (2012) ,

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 111

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erive a new model for the crosstalk between BR and GA signalling

athways, and analyse the influence of different interaction mech-

nisms and changes in the dynamics of one of the signalling path-

ays on the qualitative and quantitative behaviours of the coupled

ystem. We also analyse the influence of spatial heterogeneity of

he signalling processes on the interactions between the BR and

A signalling pathways. We summarise and discuss our results in

ection 7 .

. Biological background

The mathematical modelling and analysis of the BR signalling

athway and of the interactions between the BR and GA signalling

athways is the main aim of this paper. In this section we present

n overview of the BR and GA signalling pathways, and the inter-

ctions between them.

.1. The BR signalling pathway

The signalling process starts at the cell plasma-membrane with

he perception of BR by the receptor BRASSINOSTEROID INSENSI-

IVE1 (BRI1) ( Li and Chory, 1997 ). Upon BR binding to BRI1, two

ain events then take place, association of BRI1 to a co-receptor,

RI1-ASSOCIATED RECEPTOR KINASE1 (BAK1), and dissociation of

he inhibitor protein BRI1 KINASE INHIBITOR1 (BKI1). This triggers

transphosphorylation cascade between BRI1 and BAK1, leading

urther to phosphorylation of BRASSINOSTEROID-SIGNALLING KI-

ASE1 (BSK1), another membrane-bound kinase. Next, BSK1 phos-

horylates the protein phosphatase BRI1-SUPPRESSOR1 (BSU1),

hich dephosphorylates a protein kinase BRASSINOSTEROID IN-

ENSITIVE2 (BIN2), eventually leading to its degradation ( Ryu et al.,

010 ). In the absence of BR, phosphorylated BIN2 has a role

n phosphorylating the two transcription factors, BRASSINAZOLE

ESISTANT1 (BZR1) and BRI1-EMS-SUPPRESSOR1 (BES1) ( Li and

eng, 2005 ), also known as BZR2 (for the purposes of this paper

t is unnecessary to distinguish between the two, so we refer to

hem jointly as BZR). When phosphorylated, BZR is less stable and

hus more unlikely to activate or repress any of the 10 0 0s of genes

ssociated with BR signalling ( Ryu et al., 2007 ), it is also thought

hat association of phosphorylated BZR to a 14-3-3 protein inhibits

ts entry to the nucleus. PROTEIN PHOSPHOTASE 2A (PP2A) is re-

ponsible for the de-phosphorylation of BZR, which allows its entry

nto the nucleus and then its activation of BR responsive genes.

BZR functions as a repressor of certain genes associated with

he biosynthesis of BR, notably for example CONSTITUTIVE PHO-

OMORPHOGENESIS AND DWARFISM (CPD), DWARF4 (DWF4),

OTUNDIFOLIA3 (ROT3) and BRASSINOSTEROID-6-OXIDASE 1

BR6ox1) ( Tanaka et al., 2005 ). That is, active, de-phosphorylated

ZR inhibits production of BR. So, high levels of BR cause low lev-

ls of phosphorylated BZR, leading to inhibition of BR biosynthesis

nd decreasing levels of BR. Conversely, low levels of BR lead to

igh levels of phosphorylated BZR and activation of BR biosyn-

hesis, increasing the levels of BR. This completes the negative

eedback loop of the BR signalling pathway.

.2. The GA signalling pathway

Gibberellin Signalling is achieved by enhancing the degrada-

ion of DELLA proteins, which influence the expression of GA-

esponsive genes ( Achard and Genschik, 2009 ). GA molecules are

erceived by the GA receptor, GIBBERELLIN INSENSITIVE DWARF1

GID1), a nuclear-localised protein ( Ueguchi-Tanaka et al., 2005 ).

nalysis of GID1’s structure revealed that it has a GA-binding

ocket, with a flexible extension adjacent ( Shimada et al., 2008 ).

hen GA binds to GID1, this extension undergoes conforma-

ional change, and covers the GA-binding pocket. When closed,

he upper surface of this lid binds to DELLA proteins to form the

A.GID1.DELLA complex. The formation of the GA.GID1.DELLA com-

lex enhances the degradation of DELLA proteins by mediating

roteasome-dependent destabilization of DELLA proteins.

The GA signalling pathway exhibits negative feedback due to

he influence of DELLAs on the expression of several genes which

ode components of the signalling pathway ( Yamaguchi, 2008 ).

irst, DELLA activates the GID1-encoding gene, leading to an in-

rease in the translation of the GID1 protein. This means that in

he absence of DELLAs, GID1 concentration also decreases which

ill slow down the proteasome-induced DELLA degradation, and

hat an abundance of DELLA leads to the production of more

ID1 and enhances the DELLA degradation. Next, DELLA activates

he transcription of genes encoding the enzymes GA 20-oxidase

GA20ox) and GA 3-oxidase (GA3ox). These enzymes catalyse sev-

ral reaction steps in the GA biosynthesis pathway, meaning an

bundance of DELLA increases both GA and GID1, leading to degra-

ation of DELLA. Lastly, DELLA represses its own gene transcrip-

ion.

.3. Crosstalk between the BR and GA signalling pathways

The interaction of BRs and GAs has been receiving much atten-

ion, due to their shared nature as critical plant growth regulators,

ombined with the fact that they share many overlapping func-

ions such as regulation of cell elongation ( Catterou et al., 2001;

beda Tomás et al., 2009 ) and plant responses to abiotic stress

Ahammed et al., 2015; Colebrook et al., 2014 ). However despite the

nterest, the exact mechanisms of these interactions have remained

argely unclear, save from the fact that they control expression of

everal genes ( Bouquin et al., 2001 ). There has been much evidence

hat the signalling processes of BR and GA converge at the level

f BZR and DELLA interaction. Direct crosstalk in this fashion was

hown in Li et al. (2012) , where it was shown that overexpression

f DELLA proteins reduced both the abundance and transcriptional

ctivity of BZR. This was found to be due to the formation of a

omplex of DELLA and BZR, which removed BZR’s transcriptional

bility. There is also evidence of BRs regulating the biosynthesis of

As, the so called “GA Synthesis” model of BR-GA crosstalk. Two

ain proposals have been made for the existence of this type of

nteraction. The effects of BR mutants on GA synthesis were ex-

mined in Tong et al. (2014) , and it was concluded that BZR en-

ances GA synthesis by activating synthesis of the GA3ox enzyme.

n contrast to this the findings in Unterholzner et al. (2015) de-

cribe a much larger role for BZR in regulating GA synthesis. They

rovide evidence for a model where BZR activates the synthesis

f the GA20ox enzyme, in addition to the effects described in

ong et al. (2014) . Thus BZR would influence the biosynthesis of

A in exactly the same manner as DELLAs for other interactions,

owever the significance of this mechanism of crosstalk is not yet

stablished ( Ross and Quittenden, 2016; Tong and Chu, 2016; Un-

erholzner et al., 2016 ).

. Derivation of a mathematical model for the BR signalling

athway

In this section we derive a mathematical model of the BR sig-

alling pathway. In order to build a simple, yet sufficiently ac-

urate and efficient model incorporating BR biosynthesis negative

eedback, we first construct a reduced reaction schematic that de-

cribes the pathway mechanism. This reduction is achieved via

simplification of two principal parts of the signalling pathway:

he complex BR biosynthesis network, and the cytoplasm local-

zed phosphorylation cascade. Hence, we build a model focussing

n three key components, hormone (BR), inhibitor (BKI1) and tran-

cription factor (BZR) ( Fig. 1 ).

112 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

Fig. 1. Reaction schematic of the reduced BR signalling pathway.

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In the mathematical model we consider the binding of free

BR molecules to the BRI1 receptors leading to the dissociation of

BKI1. This is modelled as an almost instantaneous reaction, with

BR + BRI1.BKI1 interacting and resulting into BR.BRI1 + BKI1. In or-

der to model the effects of the signalling cascade induced by the

membrane-bound receptors and subsequent effects on the phos-

phorylation state of BZR, we assume that the effects of active re-

ceptors in triggering the cascade may be approximated by the

free BKI1 that is released upon this activation. We further assume

that the free BKI1 catalyses dephosphorylation of BZR-p and desta-

bilises the BIN2 proteins, thus reducing the phosphorylation of

BZR.

We denote by b the concentration of hormone BR, by k the con-

centration of inhibitor BKI1, by r k the concentration of receptor-

inhibitor complex BRI1.BKI1, by r b the concentration of receptor-

hormone complex BR.BRI1, by z the concentration of (dephospho-

rylated) transcription factor BZR, and by z p the concentration of

(phosphorylated) transcription factor BZR-p. Then assuming spatial

homogeneity of the signalling processes, the interactions between

b, k, r k , r b , z , and z p are described by the system of six ordinary

differential equations

db

dt = βk r b k − βb r k b +

αb

1 + (θb z) h b − μb b,

dk

dt = βb r k b − βk r b k,

dr k dt

= βk r b k − βb r k b,

dr b dt

= βb r k b − βk r b k,

dz

d t = δz z p k − ρz

z

1 + (θz k ) h z ,

dz p

dt = −δz z p k + ρz

z

1 + (θz k ) h z . (1)

Here βb is the binding rate of b to r k , and βk is the binding

rate of k to r b . We model this as only two reactions by assuming

that when either BR or BKI1 are bound to BRI1 either dissociates

sufficiently fast that the levels of the BR.BRI1.BKI1 remains roughly

zero. We assume the reaction to occur in some finite closed vol-

ume, so the loss of BR is only described by the degradation coeffi-

cient μb .

The phosphorylation state of BZR is modelled as being depen-

dent on the levels of free BKI1. We justify this by noting that upon

signalling activation, the receptor phosphorylates BSU1 which gov-

erns the phosphorylation of BZR via BIN2. Thus since BKI1 is also

released upon BR binding, we may model these effects by assum-

ing that free BKI1 activates (or catalyses) the dephosphorylation of

BZR-p at rate δz . In BKI1’s absence BZR is phosphorylated at rate

ρz , and when BKI1 is present it inhibits the phosphorylation of BZR

such that when k = 1 /θz , the rate of phosphorylation is halved.

Finally we model BR biosynthesis as being directly inhibited by

BZR. BZR represses the expression of several genes encoding en-

zymes, namely CPD, DWF4, ROT3 and BR6ox1, that are required for

the conversion of many of the precursors involved in BR biosynthe-

sis. Hence we use a Hill function with exponent h > 1 to model

b

he cumulative inhibitory effect of BZR on these genes. We esti-

ate the parameter h b by considering the detailed BR biosynthesis

athway(s) presented in Chung and Choe (2013) . The BR-mediated

nzymes that are involved in the biosynthesis are CPD, DWF4,

OT3, and BR6ox1 which mediate four, five, six, and five steps in

he biosynthetic pathway respectively. Since BR biosynthetic en-

ymes act multiplicatively at different steps of the reaction net-

ork, the expressions modelling their actions can be approximated

y the product of these expressions, which would mean that the

xponents of these functions would be summed, thus we consider

b = 20 .

The model equations (1) imply that the total concentrations of

KI1, BRI1 and BZR are conserved, thus we consider k + r k = K tot ,

b + r k = R tot , and z + z p = Z tot , and derive a reduced model:

db

dt = βk (R tot − K tot + k ) k − βb (K tot − k ) b +

αb

1 + (θb z) h b − μb b,

dk

dt = βb (K tot − k ) b − βk (R tot − K tot + k ) k,

dz

dt = δz (Z tot − z) k − ρz

z

1 + (θz k ) h z . (2)

Various values for R tot were reported in van Esse et al. (2011) ,

nd we chose the value for WT seedling roots. We further assume

hat K tot = R tot in order that the receptor should have the ability

o be completely inactive, but not be saturated by BKI1. We also

se physiological values reported in the literature in order to write

k and μb in terms of other parameters, for full calculations see

ppendix A . As such we are left with 8 parameters for which we

ave no direct estimate, namely βb , αb , θb , δz , Z tot , ρz , θ z and

z . These parameters were estimated indirectly by validating the

umerical solutions of the mathematical model (2) against experi-

ental results, using numerical optimisation algorithms.

By deriving the steady state concentration of BR, denoted [ BR ] 0 ,

rom the level of endogenous 24-epiBL reported in Wang et al.

2014b ), we were able to write the rate of BR degradation μb in

erms of αb , θb , Z tot , δz , ρz , θ z , h z and h b as follows

b =

αb

[ BR ] 0

(1 +

(θb

Z tot δz [ BKI1] 0 (1+(θz [ BKI1] 0 ) h z ) ρz + δz [ BKI1] 0 (1+(θz [ BKI1] 0 ) h z )

)h b ) . (3)

We can write βk in terms of βb and other known parameters

n two ways. First, using [ BR ] 0 in conjunction with the dissociation

onstant of BR.BRI1, denoted K d , reported in Wang et al. (2001) we

an estimate the steady state concentration of BKI1, denoted

BKI 1] 0 , and write βk in terms of [ BR ] 0 , [ BKI 1] 0 , K tot , R tot , and βb

s follows

k =

(K tot − [ BKI1] 0 )[ BR ] 0 (R tot − K tot + [ BK I1] 0 )[ BK I1] 0

βb , (4)

y assuming that the dissociation constant for BR.BRI1 depends on

he steady state concentrations of BR, BR.BKI1 and BR.BRI1. For the

econd expression we considered the dissociation of both BR and

KI1 from BR.BRI1.BKI1, using the value for the dissociation con-

tant of BKI1, denoted K m

, reported in Wang et al. (2014a) , as well

s K d , to directly write βk in terms of βb as

k =

K d

K m

βb . (5)

These two constraints (4) and (5) on βk correspond to two dif-

erent mechanisms for the interactions between BR, BRI1 and BKI1.

n (4) binding of BR to BRI1.BKI1 causes instantaneous dissocia-

ion of BKI1 and formation of BR.BRI1, likewise binding of BKI1

o BR.BRI1 causes instantaneous dissociation of BR and formation

f BRI1.BKI1. In (5) binding of BR to BRI1.BKI1 or binding of BKI1

o BR.BRI1 leads to the formation of BR.BRI1.BKI1, which may then

issociate into either BKI1 and BR.BRI1, or BR and BRI1.BKI1. Model

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 113

Fig. 2. BR biosynthetic gene expression calculated from the numerical solutions of

model (2) , plotted against experimental data from Tanaka et al. (2005) . For the WT

plants grown under control conditions the parameters given in Table 1 were used,

parameter sets for other cases can be found in Table A.6 .

Fig. 3. BR biosynthetic gene expression calculated from the numerical solutions of

model (2) , plotted against experimental data from Tanaka et al. (2005) . For the WT

plants grown under control conditions the parameters given in Table 2 were used.

(

(

t

e

b

i

p

p

a

i

o

o

g

f

i

d

w

a

c

t

t

w

t

0

s

v

m

n

t

t

d

B

fl

t

w

a

a

m

e

t

T

t

b

t

d

a

o

u

m

w

s

c

a

t

o

T

c

w

w

w

fi

w

e

s

3

v

4

s

y

t

a

2) was fitted to experimental data using both conditions (4) and

5) in order to compare their effects, see Figs. 2 and 3 .

The experimental data from Tanaka et al. (2005) , used to de-

ermine model parameters, give the relative BR biosynthetic gene

xpression of CPD, DWF4, ROT3, and BR6ox1, and were measured

y RT-PCR analysis, then converted to give relative values with the

nitial values equal to one. Values were measured for three inde-

endent experiments (three replicates), and the data presented by

oints and (where available) error bars correspond to the mean

nd standard error respectively. Gene expression was measured

n both Wild-Type (WT) and bri1-401 mutant (where perception

f BR by BRI1 is inhibited) plants, and this was accounted for in

ur parameter estimation by assuming that the parameter βb was

reater for the WT than for the mutant. βk was allowed to vary

reely for the mutant case since constraints (4) and (5) are not def-

nitely valid in this case. Both of these phenotypes were grown un-

er control conditions as well as two other cases: one where plants

ere grown in a medium containing 5μM of Brassinazole (BRZ),

BR-specific biosynthesis inhibitor, and one grown in a medium

ontaining 0.1 μM of Brassinolide (BL) having first been grown in

he medium containing 5 μM BRZ for two days. Data comparing

he control case with the case of growth in the 5 μM BRZ medium

ere recorded for five days. Data comparing further growth after

wo days of the BRZ medium case with the case of addition of

.1 μM BL to the medium were recorded for further 24 h, and as

uch for these cases the initial conditions were taken to be the

alues of the numerical solution to the model for the 5 μM BRZ

edium at time t = 2 days. For the control conditions we made

o amendments to the model, for the case of plants growing in

he BRZ-medium we imposed bounds upon the parameters such

hat αb should be smaller in this case since addition of BRZ re-

uces the biosynthesis of BR. In order to examine the case where

L was added to the growth medium, an extra term governing in-

ux of exogenous BL and efflux of endogenous BR was added to

he equation describing BR dynamics

db

dt = βk (R tot − K tot + k ) k − βb (K tot − k ) b +

αb

1 + (θb z) h b

−μb b + φb (ω b − b) ,

here φb is the relative permeability of the cell membrane to BR

nd was one of the optimised parameters for the relevant cases,

nd ω b is assumed to be 0.1 μM in accordance with the experi-

ental procedure.

Optimisation was achieved by comparing the biosynthetic

xpression defined by the numerical solutions of model (2) ,

he term 1 / (1 + (θb z) h b ) , with experimental data presented in

anaka et al. (2005) . In order to compare experimental data with

he output from our model we first normalised the simulation data

y their initial values such that they took values comparable to

he experimental data. We then took the mean of the four gene

ata sets, weighted by the number of times the respective proteins

ppear in the biosynthetic pathway. The optimisation was carried

ut in Python using the curve_fit function in the SciPy mod-

le ( Jones et al., 2001 ). curve_fit applies nonlinear least squares

inimization using the trust region reflective algorithm as default,

ith a default tolerance of 10 −8 . The model was fitted to the data

et for each case sequentially, starting with the WT under control

onditions since this data set was the largest. The parameters μb

nd βk were replaced by the expressions (3) and (4) or (5) , respec-

ively, for the WT data under control conditions since this is the

nly case where such parameter constraints are definitely valid.

he parameters generated from the fitting for WT under control

onditions were then used as the initial guesses for all other cases,

here μb and βk were also allowed to be fitted. Parameters that

ere not expected to vary under the different growth conditions

ere allowed very small variations to account for error in the first

tting, whereas parameters that were expected to vary had much

ider bounds.

Numerical simulations of model (2) using the optimised param-

ters, given in Tables 1 and 2 for βk determined by (4) and (5) re-

pectively, are plotted against the experimental data in Fig. 2 and

and show good agreement with experimental data, having R 2

alues of 0.89 and 0.92 respectively.

. Qualitative analysis of the mathematical model for the BR

ignalling pathway

In this section we consider well-posedness and qualitative anal-

sis of model (2) . We start by non-dimensionalising our model,

ransforming the variables as t =

1 μb

t , b =

αb μb

b , k = K tot k , z = Z tot z ,

nd introducing the dimensionless parameters

114 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

Table 1

The model parameters associated with the BR signalling pathway model (2) for WT when fitted using expression (4) for βk , together

with their sources (Parameter set 1).

Constant Value Units Source Constant Value Units Source

αb 0.27 μM min −1 fit θ z 3.95 μM

−1 fit

βb 8.33 μM

−1 min −1 fit h b 20 Chung and Choe (2013)

βk 2.73 μM

−1 min −1 (4) h z 6 fit

ρz 1 . 33 × 10 −4 min −1 fit K tot 6 . 2 × 10 −2 μM fit

δz 1 . 02 × 10 −3 μM

−1 min −1 fit R tot 6 . 2 × 10 −2 μM van Esse et al. (2011)

μb 3.58 min −1 (3) Z tot 2.65 μM fit

θ b 1.96 μM

−1 fit φb 7.06 min −1 fit

ω b 0.1 μM Tanaka et al. (2005)

Table 2

The model parameters associated with the BR signalling pathway model (2) for WT when fitted using expression (5) for βk , together

with their sources (Parameter set 2).

Constant Value Units Source Constant Value Units Source

αb 0.27 μM min −1 fit θ z 3.95 μM

−1 fit

βb 8.33 μM

−1 min −1 fit h b 20 Chung and Choe (2013)

βk 2 . 18 × 10 −2 μM

−1 min −1 (5) h z 6 fit

ρz 4 . 26 × 10 −4 min −1 fit K tot 6 . 2 × 10 −2 μM fit

δz 1 . 75 × 10 −3 μM

−1 min −1 fit R tot 6 . 2 × 10 −2 μM van Esse et al. (2011)

μb 3.68 min −1 (3) Z tot 2.68 μM fit

θ b 2.0 μM

−1 fit φb 160.93 min −1 fit

ω b 0.1 μM Tanaka et al. (2005)

P

o

a

g

a

s

β

a

c

t

o

f

t

i

u

4

i

s

a

l

m

s

s

t

t

c

c

d

r

a

c

1

βk =

βk K

2 tot

αb

, βb =

βb K tot

μb

, κ =

R tot

K tot , θb = θb Z tot ,

ε =

αb

K tot μb

, δz =

δz K tot

μb

, ρz =

ρz

μb

, θz = θz K tot ,

which yields the system (neglecting s)

db

dt = f 1 (b, k, z) = βk (κ − 1 + k ) k −βb (1 − k ) b +

1

1 + (θb z) h b − b,

dk

dt = f 2 (b, k, z) =

1

ε( βb (1 − k ) b − βk (κ − 1 + k ) k ) ,

dz

d t = f 3 (b, k, z) = δz (1 − z) k − ρz

z

1 + (θz k ) h z . (6)

For simplicity of presentation we denote by P ⊂ [1 , ∞ ) × R

7 + ×N

2 the parameter space for system (6) , where for each p ∈ P , p =(κ, βk , βb , θb , ε, δz , ρz , θz , h b , h z ) . We assume that κ has a minimum

value of 1 since κ < 1 would imply saturation of receptor by in-

hibitor (i.e. K tot > R tot ), leading to BR signalling being permanently

switched on.

Theorem 1. The system (6) has a unique, global solution ( b,

k, z ) ∈ C 1 ([0, ∞ )) for any initial value (b 0 , k 0 , z 0 ) ∈ [0 , 1 + βk κ] ×[0 , 1] 2 , and (b(t) , k (t) , z(t)) ∈ [0 , 1 + βk κ] × [0 , 1] 2 for all t ∈ [0, ∞ )

and any p ∈ P.

Proof. Define u = (b, k, z) T and f = ( f 1 , f 2 , f 3 ) T , and hence du

dt = f .

Since f is locally Lipschitz-continuous, the Picard-Lindelöf theo-

rem ensures local existence of a unique solution of (6) , see e.g.

( Amann, 1990 ). To obtain global existence and uniqueness we

prove boundedness of solutions by demonstrating the existence of

a positive-invariant region for system (6) , i.e. showing that for a

solution u of (6) starting in M = [0 , 1 + βk κ] × [0 , 1] × [0 , 1] it will

always be contained within this region. To show that a region M

is positive invariant under the flow of system (6) , we show that

f (u ) · n (u ) ≥ 0 ∀ u ∈ ∂M, see e.g. ( Amann, 1990 ), where n is the in-

ward normal vector on ∂M , see B.1 for more details. This implies

uniform boundedness of solutions of (6) with initial values in M ,

and continuous differentiability of f ensures global existence and

uniqueness. �

Theorem 2. For any parameter set p ∈ P, there exists a unique steady

state solution ( b ∗, k ∗, z ∗) ∈ M of system (6) .

roof. Considering equations for a steady state solution ( b ∗, k ∗, z ∗)

f (6) and employing simple algebraic manipulation, k ∗ is defined

s a root of the following non-linear function

(k ∗) : = βk ( κ − 1 + k ∗) k ∗

⎛

⎝ 1 +

(

θb δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

)) h b

⎞

⎠

− βb ( 1 − k ∗) ,

nd b ∗ and z ∗ are determined as functions of k ∗, for more details

ee ( B.2 ). We may immediately see, since g(0) = −βb and g(1) =k κ(1 + (θb δz (1 + θh z

z ) / (ρz + δz (1 + θh z z )) h b ) , that g must contain

t least one root in [0, 1] for any p ∈ P , and hence system (6) must

ontain at least one steady state solution in M . In order to find

he number of roots of g(k ∗) = 0 in [0, 1], consider the derivative

f g which is positive for all p ∈ P and k ∗ ∈ [0, 1], see ( B.2 ) for the

ormula for g ′ ( k ∗). Thus g is monotonically increasing. Strict mono-

onicity of g coupled with existence of at least one root in [0, 1]

mplies that g has a unique root in [0, 1], and hence (6) has a

nique steady state in M . �

.1. Linearised stability and bifurcation analysis

To study the qualitative behaviour of solutions of mathemat-

cal model for BR signalling pathway, we performed linearised

tability analysis for system (6) and analyse the impact of vari-

tions in values of model parameters on the behaviour of so-

utions of (6) . For the parameters obtained via validation of

athematical model by experimental data, see Tables 1 and 2 ,

teady state solutions of (6) are linearly stable, with eigenvalues

(−3 . 8764 , −0 . 1508 , −0 . 0 0 09) and (−3 . 6863 , −0 . 1138 , −0 . 0 0 06) re-

pectively. Further, stability of steady state solutions is main-

ained under moderate variations of all parameters, suggesting

hat the BR homeostasis is ensured in normally functioning plant

ells. Large variations in δz , ρz , and θ z however cause qualitative

hanges in the behaviour of solutions of model (2) and can in-

uce oscillatory behaviour, but only in the case when all other pa-

ameters are as in Table 1 and not as in Table 2 . We consider δz

nd ρz as bifurcation parameters because these parameters directly

orrespond to processes in the BR signalling pathway, whereas

/ (1 + (θz k ) h z ) is only an approximation for the dynamics of the

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 115

Fig. 4. Critical dimensional values δz and ρz , at which the complex conjugate pair

of eigenvalues λ2 , λ3 of (B.1) are purely imaginary, form a closed curve L . Note the

third eigenvalue λ1 is always negative. The system (6) exhibits oscillatory behaviour

when values of ( δz , ρz ) are located within the region bounded by curve. For δz = 6

fixed, the eigenvalues cross the imaginary axis at ρ(1) z = 3 . 78 and ρ(2)

z = 24 . 1 with

values of d dρz

Re (λ2 , 3 ) of 1 . 78 × 10 −3 and −4 . 53 × 10 −4 respectively. For ρz = 6 fixed,

the eigenvalues cross the imaginary axis at δ(1) z = 3 . 14 and δ(2)

z = 8 . 87 with values

of d dδz

Re (λ2 , 3 ) of 2 . 73 × 10 −3 and −1 . 33 × 10 −3 respectively.

c

c

B

d

t

o

I

v

v

m

g

fi

t

ρ

T

g

P

p

e

p

s

s

a

B

t

t

c

fi

s

s

w

e

t

c

H

s

F

p

v

t

(

h

o

n

a

J

e

F

w

d

w

a

a

o

t

δ

ρ

a

t

f

H

i

s

l

D

fi

(

d

s

5

t

p

o

b

n

r

l

d

c

t

o

i

t

r

m

e

a

d

a

o

ytoplasmic phosphorylation cascade. Biologically, increase of δz

ould potentially correspond to faster phosphorylation of BSU1 by

AK1, or decrease of δz corresponding to reduced action of PP2A in

ephosphorylating BZR. Further, decrease of ρz could correspond

o BIN2-deficient or insensitive mutants e.g. bes1-D, and increase

f ρz could correspond to BIN2-overexpressing mutants, e.g. bin2.

n the bifurcation analysis of system (2) we considered increased

alue for the dimensional parameter θ z compared to the standard

alue, Table 1 (i.e. θz = 41 . 2 μM

−1 ), which was essential to deter-

ine the region for parameters δz and ρz where system (6) under-

oes bifurcation. For the value of θz = 3 . 95 μM

−1 obtained through

tting model solutions to experimental data, the steady state solu-

ion of (6) is linearly stable for a wide range of values of δz and

z , i.e. δz , ρz ∈ (0, 50).

heorem 3. As δz and ρz are continuously varied, system (6) under-

oes a Hopf bifurcation.

roof. We performed linearised stability analysis to determine the

arameter subspace for which the stationary solution of (6) is lin-

arly stable, as well as the range of parameters for which we have

eriodic solutions for the model (6) .

Using the Jacobian (B.1) of system (6) , evaluated at the steady-

tate ( b ∗, k ∗, z ∗), we calculate the characteristic equation for the

ystem to be a cubic polynomial of the form λ3 + a 2 λ2 + a 1 λ +

0 = 0 , where a 2 , a 1 and a 0 are all positive, real constants (see

.3 ). Since the characteristic equation is a cubic polynomial we ob-

ain that there are only 2 possible sets of eigenvalues, either that

hey are all real or that there is one real eigenvalue λ1 and two

omplex conjugate eigenvalues λ2 and λ3 . Further, since all coef-

cients have the same sign any real eigenvalue must be negative,

pecifically zero cannot be an eigenvalue of J for any parameters of

ystem (6) in P . Together these two facts tell us that in the case

here the eigenvalues are all real, or that the complex conjugate

igenvalues have negative real part the steady state is stable, and

hat there is a possible bifurcation point when the two complex

onjugate eigenvalues cross the imaginary axis, corresponding to a

opf bifurcation.

We showed numerically that for a closed loop L in ( δz , ρz ) the

ystem has complex conjugate eigenvalues with zero real parts,

ig. 4 , and in the region D enclosed by the loop L the com-

lex conjugate eigenvalues have positive real part, whereas for

(δz , ρz ) ∈ (0 , 50) 2 \ D the real part of the complex conjugate eigen-

alues is negative. Hence the points of the loop L correspond to

he bifurcation points where the stability of stationary solutions of

6) changes. We also showed that at such points the eigenvalues

ave non-zero imaginary parts and hence do not pass through the

rigin, Fig. 5 a), which supports the proof of the fact that zero can-

ot be an eigenvalue of J , presented above. For this we designed

scheme in MATLAB to calculate the eigenvalues of the Jacobian

in (B.1) , for values of ( δz , ρz ) ∈ (0, 50) 2 , with dimensional param-

ter values θz = 41 . 2 μM

−1 and all other parameters as in Table 1 .

or each δz ∈ (0, 50), we determined the values of ρz ∈ (0, 50) for

hich J has a pair of non-zero purely imaginary eigenvalues. The

erivatives of the real part of the eigenvalues w.r.t. both δz and ρz

ere also calculated numerically, Fig. 5 b). The values of d dδz

Re (λ2 , 3 )

nd

d dρz

Re (λ2 , 3 ) at the critical points are non-zero, apart from ex-

ctly four points on the curve in Fig. 5 b), where the derivative w.r.t.

ne of the parameters will be zero. Those four points correspond

o the points where δz or ρz are at their extreme values, i.e. when

z takes an extreme value we have that d dρz

Re (λ2 , 3 ) = 0 , and when

z takes an extreme value that d dδz

Re (λ2 , 3 ) = 0 . When δz is fixed

t an extreme point, varying ρz will not cause the point to enter

he region bounded by the curve and will not correspond to a bi-

urcation point w.r.t. ρz , similar for ρz fixed at an extreme point.

ence zero derivative at this point does not break the transversal-

ty condition. Thus, in conjunction with Theorems 1 and 2 we have

hown that system (6) satisfies all conditions for the existence of a

ocal Hopf bifurcation ( Hassard et al., 1981 ).

Therefore, for all parameter sets such that (δz , ρz ) ∈ (0 , 50) 2 \ , θz = 41 . 2 μM

−1 , and all other dimensional parameters are de-

ned as in Table 1 , we have that the steady state solution of system

6) is linearly stable. At the points (δz , ρz ) ∈ L the system (6) un-

ergoes a Hopf bifurcation, and for (δz , ρz ) ∈ D we have periodic

olutions for the BR signalling pathway model. �

. Spatially heterogeneous model for the BR signalling pathway

The BR signalling pathway is a process which has distinct func-

ions at different spatial locations in the cell ( Fig. 6 ). Thus it is im-

ortant to extend the ODE model (1) and analyse the dependence

f the dynamics of the pathway components on the spatial distri-

ution.

We consider a spatially heterogeneous model for the BR sig-

alling pathway in the one-dimensional domain c = (0 , l c ) rep-

esenting a part of the plant cell cytoplasm, where l c denotes the

ength of the cell segment we consider. The boundaries of c are

enoted by �n modelling the cell nucleus, and �c representing the

ell membrane ( Fig. 7 ).

We assume the diffusion of BR ( b ), BKI1 ( k ) and BZR-p ( z p ) in

he cytoplasm:

∂ t b = D b ∂ 2 x b − μb b

∂ t k = D k ∂ 2 x k

∂ t z p = D z ∂ 2 x z p

⎫ ⎬

⎭

in c . (7)

The only reaction that takes place in c is the degradation

f BR since we assume that the phosphorylation status of BZR

s modulated only in the nucleus. The dynamics occurring on

he plasma membrane �c are the interactions between b, k , and

eceptor BRI1 ( r k , r b ). Since we assume that the receptors are

embrane-bound a system of ODEs, similar to the corresponding

quations in system (1) , is considered to model the dynamics of r k nd r b . The effect of the interactions between b, k, r k , and r b on the

ynamics of b and k is defined by Robin boundary conditions for b

nd k on �c . Finally, we assume that the BZR-p cannot diffuse out

f the cell, which we model by a zero-flux boundary condition on

116 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

Fig. 5. Numerical verification of the existence of a Hopf bifurcation. a) at the critical values, Im ( λ2, 3 ) are non-zero. b) derivatives of Re ( λ2, 3 ) w.r.t. both δz and ρz (green

dots and magenta crosses respectively). Values pass through zero at the points of the curve in a) where δz or ρz take their extrema. At such points there is a bifurcation

only in one of the parameters, the parameter for which Re ( λ2, 3 ) has non-zero derivative. (For interpretation of the references to colour in this figure legend, the reader is

referred to the web version of this article.)

Fig. 6. Diagram of the spatial heterogeneity considered for the model of the BR sig-

nalling pathway (7) - (9) . BR (red circles), BKI1 (blue squares) and BZR-p (yellow dia-

monds with black dots) diffuse freely in the cytoplasm, where BR is also degraded.

At the membrane, both BR and BKI1 are perceived by BRI1 (black y-shapes) and

form complexes with it. In the nucleus (brown) BKI1 activates dephosphorylation

of BZR-p to BZR (yellow diamonds), and inhibits phosphorylation of BZR to BZR-p.

BR is synthesised in the endoplasmic reticulum (ER, grey crescent) which is contin-

uous with the nuclear membrane. (For interpretation of the references to colour in

this figure legend, the reader is referred to the web version of this article.)

Fig. 7. A diagram of the one-dimensional domain in which system (7) - (9) was

solved. c = (0 , l c ) represents the cytoplasm, �c the plasma membrane, and �n the

nucleus.

v

t

(

θb

m

S

f

l

f

s

t

p

t

h

(

i

c

t

m

6

t

t

p

p

e

s

(

t

t

T

e

t

6

e

O

G

G

G

c

p

a

�c . Thus on �c we have

−D b ∂ x b = βk r b k − βb r k b −D k ∂ x k = βb r k b − βk r b k −D z ∂ x z p = 0

d r k dt

= βk r b k − βb r k b d r b dt

= βb r k b − βk r b k

⎫ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎭

on �c . (8)

In the nucleus �n we consider both the phosphorylation and

dephosphorylation of BZR. Although the exact subcellular location

of BR biosynthesis has not been experimentally demonstrated, the

likely location is the endoplasmic reticulum ( Shimada et al., 2001 ).

The endoplasmic reticulum is continuous with the nuclear mem-

brane, hence we model the BR biosynthesis as occurring on �n .

We assume that BKI1 cannot enter the nucleus and consider zero-

flux boundary conditions for k on �n .

D b ∂ x b =

˜ αb

1+( θb z ) h b

D k ∂ x k = 0

D z ∂ x z p = − ˜ δz z p k + ρz ˜ z

1+(θz k ) h z d z d t

= δz z p k − ρz ˜ z

1+(θz k ) h z

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

on �n . (9)

Since ˜ r k , ˜ r b and ˜ z are confined to the boundary, they have

units of mol / m

2 . Thus we have the following scaled relationships

between variables ˜ r , ˜ r and ˜ z in (7) - (9) and the corresponding

k b

ariables r k , r b and z in (1) : ˜ r k = l c r k , ˜ r b = l c r b , ˜ z = l c z. In order

o preserve the balance of units some parameters from model

1) also had to be rescaled, specifically ˜ αb = l c αb ,˜ δz = l c δz and

˜ b = θb /l c . We estimated the diffusion constant D b = 60 μm

2 min

−1

y taking the value reported for Progesterone (a steroidal hor-

one with a similar structure to BL) in physiological solution from

ieminska et al. (1997) . We also took D k = D z = 0 . 125 μm

2 min

−1

or the diffusion constant for proteins from Sturrock et al. (2011) .

c was taken to be 7.43 μm from measurements of root cell sizes

rom unpublished data. All other parameters were taken to be the

ame as in Table 1 or Table 2 .

Model (7) - (9) was solved numerically to analyse the changes of

he model solutions due to the spatial heterogeneity of signalling

rocesses. In the fitted parameter regimes, averaged solutions of

he model (7) - (9) have similar behaviour to solutions of (2) ( Fig. 8 ),

owever a distinct spatial distribution of BR is characteristic for

7) - (9) , see Fig. 9 . In the oscillatory parameter regime discussed

n Section 4.1 , the PDE-ODE model (7) - (9) was found to have in-

reased amplitude and period of oscillations, see Fig. 8 . For solu-

ions of model (7) - (9) we also observe oscillatory behaviour for a

uch wider range of values of δz and ρz than for model (2) .

. Derivation of a mathematical model for crosstalk between

he BR and GA signalling pathways

Here we consider a mathematical model for the crosstalk be-

ween the BR and GA signalling pathways. BRs and GAs can

lay similar independent roles in development both having im-

ortant roles in growth ( Clouse and Sasse, 1998; Úbeda Tomás

t al., 2009 ), as well as acting together via shared gene expres-

ion, and through interactions between their signalling pathways

Bouquin et al., 2001; Tanaka et al., 2003 ). We aim to use our model

o analyse the three different mechanisms of crosstalk between

he BR and GA signalling pathways proposed in Li and He (2013) ,

ong et al. (2014) and Unterholzner et al. (2015) and to attempt to

stablish which mechanism has the most significant influence on

he dynamics of the pathways.

.1. Mathematical modelling of GA signalling

A detailed model of the GA signalling pathway was derived and

xamined in Middleton et al. (2012) , consisting of a system of 21

DEs and 42 parameters. It considers both the interaction of the

A 4 , GID1 and DELLA, and the GA 4 biosynthesis pathway, where

A 12 is converted to GA 15 is converted to GA 24 is converted to

A 9 , all catalysed by enzyme GA20ox, and GA 9 is converted to GA 4 ,

atalysed by GA3ox.

Since a model for crosstalk must include variables for both

athways involved, as well as potentially introducing new vari-

bles specific to their interactions, a system of ODEs describing

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 117

Fig. 8. Comparison between the dynamics of BR for the ODE model (6) , and the PDE-ODE model (7) - (9) for various parameter values. For comparison, the solutions to

the PDE-ODE model have been averaged over the space, and the initial conditions are such that they are equal for the ODEs and the averaged PDEs. a) Both models

were solved with the fitted parameters (parameters in Tables 1 and 2 correspond to set 1 and set 2 respectively), and show similar behaviour. b) Models were solved for

δz = 1 . 02 × 10 −2 , ρz = 1 . 33 × 10 −3 , θz = 41 . 2 and all other parameters as in Tables 1 and 2 respectively. The ODE model tends quickly to steady state, whereas the PDE-ODE

model exhibits damped oscillations. c) Models were solved for δz = ρz = 4 , θz = 41 . 2 , and all other parameters as in Table 1 . Both systems exhibit periodic solutions, but the

PDE-ODE model has a much reduced frequency and much increased amplitude as compared to the ODE model. d) δz = 14 , ρz = 35 , θz = 41 . 2 , and all other parameters as

in Table 1 , ODE model has moved outside of the oscillatory parameter regime, however the same is not true for the PDE-ODE model, which continues to have oscillatory

behaviour for values of δz , ρz in excess of 10 0 0.

Fig. 9. The spatial distribution of BR in the cell segment c , solution of (7) - (9) , at different times, where ‘parameter set1’ and ‘parameter set2’ correspond to parameter

values in Table 1 and Table 2 respectively. The steady state concentration of BR is greater for solutions with parameter set 1, Table 1 , than for solutions with parameter set 2,

Table 2 .

s

r

M

t

p

c

a

e

a

b

m

t

o

H

W

m

a

n

ϑ

a

f

G

s

uch a model may be very large. This being the case, we first

educe the size of the GA signalling model that was derived in

iddleton et al. (2012) . We reduce the model by assuming that

he dynamics of the molecules involved in the GA biosynthesis

athway are much faster compared to the dynamics of other pro-

esses involved in the signalling pathway that their levels remain

t a steady state. This enables us to write a system of algebraic

quations governing GA 12 , GA 15 , GA 24 , GA 9 , GA20ox and GA3ox,

s well as their relevant mRNAs and complexes which may then

e solved such that they may therefore be substituted out of the

ain system. These substitutions generate a new term governing

he biosynthesis of GA 4 , where biosynthesis is directly dependent

n DELLA concentration, which is then simplified to a fourth order

ill function, signifying the 4 steps in the biosynthesis pathway.

e also assume that only one configuration of DELLA.GID1 c .GA 4

ay be formed. Overall this removes 27 parameters and 13 vari-

bles from the system in Middleton et al. (2012) , and introduces a

ew term for the GA 4 biosynthesis with new parameters αg and

g . The interested reader may examine both the reduction process

s well as explicit calculations of αg and ϑg in Appendix D . Then

or r , r o g , r c g , r d , r m

and d m

, the concentrations of GID1, GID1 o .GA 4 ,

ID1 c .GA 4 , DELLA.GID1 c .GA 4 , GID1 mRNA and DELLA mRNA re-

pectively, we obtain

dr

dt = −βg rg + γg r

o g + αr r m

− μr r,

dr o g

dt = βg rg − γg r

o g + λo r c g − λc r o g ,

dr c g

dt = −λo r c g + λc r o g − βd d l r

c g + (γd + μd ) r d ,

dr d dt

= βd d l r c g − (γd + μd ) r d ,

dr m

dt = φr

(d l

d l + ϑ r − r m

),

dd m

dt = φd

(ϑ d

d l + ϑ d

− d m

), (10)

118 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

Table 3

Default parameter values used for the GA signalling model. The values of all parame-

ters except αg and ϑg were taken directly from Middleton et al. (2012) from which our

reduced model has been derived. The values for αg and ϑg were calculated using the

parameters from Middleton et al. (2012) and the expressions for αg and ϑg arising from

the model reduction, see Appendix D .

Constant Value Units Constant Value Units

βg 1.35 μM

−1 min −1 γ g 2.84 min −1

αr 19.3 μM min −1 μr 3.51 min −1

λo 0.0776 min −1 λc 0.0251 min −1

βd 10 μM min −1 γ d 0.133 min −1

μd 6.92 min −1 αd 5 . 28 × 10 −4 μM min −1

φg 1 . 061 × 10 −3 min −1 ω g 154.27 μM

αg 6 . 40 × 10 −3 μM min −1 ϑg 4 . 27 × 10 −13 μM

4

μg 0.291 μM min −1 φr 0.0457 min −1

ϑr 5 . 6 × 10 −4 μM φd 0.0708 min −1

ϑd 0.01 μM

Fig. 10. A comparison between the numerical simulations of the full model for the GA signalling pathway derived in Middleton et al. (2012) , and the reduced model (10),

(11) shows very good agreement.

Fig. 11. BR-GA crosstalk at the level of transcription factors, where BZR and DELLA

form a complex.

B

w

s

a

a

t

t

e

L

t

t

w

p

i

G

e

s

G

and for d l and g , the concentrations of DELLA and GA 4 respectively,

we have

dd l dt

= −βd d l r c g + γd r d + αd d m

,

dg

dt = φg (ω g − g) − βg rg + γg r

o g + αg

d 4 l

d 4 l

+ ϑ g

− μg g. (11)

All parameter values are taken from those for the full model

in Middleton et al. (2012) ( Table 3 ). For all components present in

both the full and the reduced model, the initial conditions are

taken to be the same as in Middleton et al. (2012) . Both the full and

reduced models were solved numerically, and these solutions are

plotted in Fig. 10 . The two models show excellent agreement being

almost indistinguishable for six terms, and only minor short-term

discrepancies for DELLA.GID1 c .GA 4 and DELLA. Having thus suc-

cessfully reduced the GA signalling pathway model while retaining

its core behaviour, we derive a model of the crosstalk between the

BR and GA signalling pathways by coupling models (2) and (10),

(11) .

6.2. The crosstalk model

We consider direct crosstalk between the BR and GA signalling

pathways. Besides shared gene expression, two main mechanisms

for BR-GA crosstalk have been suggested: direct interaction at the

level of transcription factors BZR and DELLA ( Li and He, 2013 ), and

ZR-mediated biosynthesis of GA ( Unterholzner et al., 2015 ). Here

e derive a mathematical model to compare the behaviour of the

ystem under the three different mechanisms of crosstalk (inter-

ction between BZR and DELLA, BZR-mediated biosynthesis of GA,

nd a combination of both of them) and to analyse their effects on

he BR and GA signalling processes.

The existence of an interaction between BZR and DELLA consti-

uting a crosstalk between the BR and GA signalling pathways was

stablished in Bai et al. (2012) , Gallego-Bartolomé et al. (2012) and

i et al. (2012) . Further to this, in Li and He (2013) it was found

hat the formation of a complex BZR.DELLA inhibits the transcrip-

ional activities of both BZR and DELLA ( Fig. 11 ).

A second mechanism of BR-GA crosstalk has been proposed

here BZR also has a direct influence on the GA 4 biosynthesis

athway ( Fig. 12 ). In Tong et al. (2014) it was discovered that BR

nduces the expression of GA3ox-2, leading to the accumulation of

A 1 , the most bioactive GA for rice. It was also discovered that the

xternal application of BR induces GA20ox expression in Arabidop-

is ( Lilley et al., 2013 ). Later it was shown that BRs both regulate

A20ox expression, and are required for the transcription of GA3ox

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 119

Fig. 12. Proposed mechanisms for the influence of BZR on GA biosynthesis. The

first model (green) comes from Lilley et al. (2013) , where BZR influences the ex-

pression of GA20ox, in the second model (red) from Unterholzner et al. (2015) BZR

affects both GA20ox and GA3ox expression, and in the third model (blue) from

Tong et al. (2014) BZR affects only GA3ox expression. (For interpretation of the ref-

erences to colour in this figure legend, the reader is referred to the web version of

this article.)

(

t

e

p

h

(

i

r

t

t

m

n

m

t

d

r

s

i

B

t

m

t

r

m

φ

a

l

W

w

t

m

e

f

v

B

m

a

a

c

a

o

o

e

t

p

s

w

c

t

f

p

θ

c

p

s

b

t

t

a

c

t

w

t

p

t

a

h

W

c

α

i

n

G

t

a

D

t

T

i

t

6

m

p

G

t

p

a

Unterholzner et al., 2015 ). Despite good evidence for the coexis-

ence of these two mechanisms, it is still unclear to what extent

ach mechanism operates to effect changes in BR and GA signalling

rocesses ( Ross and Quittenden, 2016; Tong and Chu, 2016; Unter-

olzner et al., 2016 ).

To derive the model for BR-GA crosstalk we coupled model

2) and model (10), (11) by adding new interaction terms describ-

ng the crosstalk mechanisms. Two terms were added, a term cor-

esponding to the interaction of BZR and DELLA, which required

he introduction of a new variable denoting the concentration of

he complex BZR.DELLA, and a term corresponding to the BZR-

ediated biosynthesis of GA. By varying the parameters of these

ew terms, we were then able to analyse the influence that each

echanism exert over the corresponding signalling processes.

BZR.DELLA complex formation is modelled as a reversible reac-

ion between BZR and DELLA. The concentration of BZR.DELLA is

enoted by z d , and the parameters βz and γ z denote the binding

ate of BZR and DELLA and the dissociation rate of BZR.DELLA, re-

pectively,

dz d dt

= βz zd l − γz z d . (12)

From this we modify system (2) to take account of the dynam-

cs of BZR.DELLA

db

dt = βk (R tot − K tot + k ) k − βb (K tot − k ) b +

αb

1 + (θb z) h b − μb b,

dk

dt = βb (K tot − k ) b − βk (R tot − K tot + k ) k,

dz

d t = δz (Z tot − z − z d ) k − ρz

z

1 + (θz k ) h z − βz zd l + γz z d . (13)

System (11) is modified by taking account of the dynamics of

ZR.DELLA, and extending the Hill function describing GA biosyn-

hesis to include BZR-mediated gene expression. For the crosstalk

odel we do not assume that exogenous GA is entering the sys-

em, hence the term φg (ω g − g) describing this process in (11) is

emoved. BZR is assumed to control gene expression in the same

anner as DELLA, but with relative activity described by φz , where

z is the ratio between the binding thresholds of DELLA and BZR

dd l dt

= −βd d l r c g + γd r d + αd d m

− βz zd l + γz z d ,

dg

dt = αg

(d l + φz z) 4

ϑ g + (d l + φz z) 4 − βg rg + γg r

o g − μg g. (14)

The values for the parameters βz , γ z governing the direct inter-

ction between DELLA and BZR are estimates from values for simi-

ar complex formation and separation from Middleton et al. (2012) .

e first assumed that the binding thresholds of BZR and DELLA

ere equal i.e. φz = 1 , and then analysed the influence of varia-

ion in φz on the dynamics of the solutions of (10) , (12) –(14) .

In order to analyse the relative effects of the different crosstalk

echanisms, we examined the behaviour of the model for differ-

nt values of βz , γ z and φz . The model was solved numerically for

our different conditions: no crosstalk ( βz , γz , φz = 0 ), crosstalk

ia BZR activated GA biosynthesis only ( βz , γz = 0 ), crosstalk via

ZR.DELLA complex formation only ( φz = 0 ), and crosstalk via both

echanisms ( Fig. 13 ). Only for variables including GA were there

ny differences when including BZR-mediated biosynthesis of GA,

nd for all other variables there was close agreement between the

ases of no crosstalk and crosstalk via biosynthesis only, and close

greement between the cases of crosstalk via complex formation

nly and crosstalk via both mechanisms. To examine the influences

f both mechanisms on the BR and GA signalling pathways, the rel-

vant parameters were varied. Variation in βz and γ z led to long-

erm behaviour changes for the components of the BR signalling

athway, and short-term changes for the components of the GA

ignalling pathway ( Fig. 14 ). Variations in φz had very little effect

ith differences in behaviour in variables containing GA for the

ases φz = 0 and φz > 0 (results omitted). From this we concluded

hat direct interaction between BZR and DELLA is the predominant

orm of crosstalk between the BR and GA signalling pathways.

In order to examine the effects of mutations in the BR signalling

athway upon BR-GA crosstalk, changes in parameters δz , ρz and

z are introduced into model (10) , (12) –(14) , Fig. 15 . Damped os-

illations in the dynamics of the components of the BR signalling

athway induced no such behaviour in the dynamics of the GA

ignalling pathway, and despite causing some changes in the early

ehaviour of the dynamics of the GA signalling pathway their long

erm behaviour remains similar. Alongside the induced perturba-

ion in the BR signalling pathway, we also varied βz and γ z to ex-

mine how crosstalk affects the dynamics of both pathways in this

ase. We found in the case that βz / γ z is large, the dynamics of

he BR signalling pathway are virtually unchanged from the cases

ith smaller values of βz / γ z , however the long-term behaviour of

he GA signalling pathway was significantly effected, with all com-

onents except GA 4 and DELLA m

having substantially different sta-

ionary solutions.

Given the observed large effect of crosstalk on the BR pathway

s compared with the GA pathway, see Figs. 13 and 14 , we analysed

ow directly altering each pathway would affect the other ( Fig. 16 ).

e modelled the overexpression of BR and GA hormones by in-

reasing parameters αb and αg . Overexpression of BR (increase of

b ) resulted in major changes in the BR signalling pathway with

ncreases in the concentrations of BR, BKI1 and BZR, but almost

egligible change in the GA signalling pathway. Overexpression of

A had more widespread effects, causing increases in the concen-

rations of BZR, GID1 o .GA 4 , GID1 c .GA 4 , DELLA.GID1 c .GA 4 and GA 4 ,

nd decreases in the concentrations of BR, BKI1, GID1 o , GID1 m

and

ELLA.BZR.

In all cases of BR-GA interactions discussed here we considered

he parameter values for the BR signalling pathway model as in

able 1 . However similar behaviours are observed also if consider-

ng the parameter values as in Table 2 , hence we do not present

he simulation results for those cases.

.3. Modelling spatial heterogeneity in the crosstalk signalling

We also considered the spatial heterogeneity in the crosstalk

odel. Here, the coupled PDE-ODE model for the BR signalling

athway (7) –(9) was modified to include the components of the

A signalling pathway. GA was assumed to be able to diffuse freely

hroughout the cell, and all other components of the GA signalling

athway were assumed to be nuclear-localized. We retain the same

ssumptions for the components of the BR signalling pathway as in

120 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

Fig. 13. Comparisons between solutions of the crosstalk model (10) , (12) –(14) ; ‘ no crosstalk ’ refers to the case where βz = γz = φz = 0 ; ‘ biosynthesis ’ refers to the case where

the only crosstalk between the BR and GA signalling pathways is BZR-induced biosynthesis of GA ( βz = γz = 0 , φz = 0); ‘ complex ’ refers to the case where the only crosstalk

between the BR and GA signalling pathways is complex formation of BZR and DELLA ( βz , γ z = 0, φz = 0 ); ‘ full ’ refers to the case where crosstalk between the BR and GA

signalling pathways can be via both mechanisms BZR-mediated GA biosynthesis and complex formation of BZR and DELLA ( βz , γ z , φz = 0).

Fig. 14. Variation of the values of βz and γ z suggests that the BR signalling pathway is more sensitive to the effects of crosstalk than the GA signalling pathway; ‘ no

crosstalk ’ corresponds to the case βz = γz = 0 , ‘ small ’ the case where βz / γ z has order of magnitude zero, i.e. βz / γ z ≈ 0.75 , for ‘ standard ’ βz / γ z has order of magnitude two,

i.e. βz / γ z ≈ 75, and for ‘ large ’ βz / γ z has order of magnitude four, i.e. βz / γ z ≈ 7500.

o

d

s

D

o

Section 5 . This led to a set of reaction-diffusion equations for BR,

BKI1, BZR-p the same as in (7) , and for GA:

∂ t g = D g ∂ 2 x g − μg g in c . (15)

We assume that receptor based interactions of BR, BKI1 and

BRI1 remain the same as in (8) , and no influx of exogenous GA

on the plasma membrane:

−D g ∂ x g = 0 on �c . (16)

The production of BR, change in phosphorylation status of BZR,

and interactions between BZR and DELLA occur in the plant cell

nucleus and are modelled as flux boundary conditions and ODEs

n the boundary �n :

D b ∂ x b =

˜ αb

1+( θb z) h b

D k ∂ x k = 0

d z d t

=

˜ δz z p k − ρz ˜ z

1+(θz k ) h z − ˜ βz z d l + γz z d

D z ∂ x z p = − ˜ δz z p k + ρz ˜ z

1+(θz k ) h z

⎫ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎭

on �n . (17)

All processes of the GA signalling pathway apart from degra-

ation of GA are localised to the nucleus and are modelled by a

ystem of ODEs for GID1 o , GA 4 .GID1 o , GA 4 .GID1 c , DELLA.GA 4 .GID1 c ,

ELLA, GID1 m

and DELLA m

, and a flux boundary condition for GA

n the boundary �n :

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 121

Fig. 15. Oscillations in the BR pathway lead to short term changes in the dynamics of the GA pathway, and increasing βz / γ z causes more long term effects on the GA

signalling pathway. In the ‘ standard ’ case all parameters are as in Tables 1 and 3 and βz = 10 μM

−1 min −1

, γz = 0 . 133 min −1

. For the ‘ BR perturbed ’ case parameters δz and ρz

have had their orders of magnitude increased by three, θz = 41 . 2 μM

−1 , and all other parameters are as in Tables 1 and 3 and βz = 10 , γz = 0 . 133 . The ‘ BR perturbed, βz / γ z

large ’ has the same parameter values as the ‘ BR perturbed ’, however here βz / γ z has order of magnitude four, i.e. βz / γ z ≈ 7500, similar to Fig. 14 .

Fig. 16. Examining the effects of hormonal overexpression on the dynamics of solutions of model (10) , (12) –(14) for the crosstalk between the BR and GA signalling pathways.

For the ‘ standard ’ case all parameters are as in Tables 1 and 3 and βz = 10 μM

−1 min −1

, γz = 0 . 133 min −1

; in the ‘ BR overexpression ’ case, αb ’s order of magnitude is increased

by one; in the ‘ GA overexpression ’ case αg ’s order of magnitude is increased by one.

122 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

g

g

l

o

g

t

t

o

e

b

p

h

e

p

w

0

a

o

w

s

i

T

o

a

d

s

o

e

o

f

t

o

r

b

t

s

o

s

n

p

(

p

w

s

S

O

t

o

p

G

a

2

(

s

t

b

b

t

s

m

T

i

c

d r dt

= −βg r g + γg r o g + ˜ αr r m

− μr r

d r o g

dt = βg r g − (γg + λc ) r o g + λo ˜ r c g

d r c g dt

= λc ˜ r o g − λo ˜ r c g − ˜ βd ˜ d l r c g + (γd + μd ) r d

d r d dt

=

˜ βd ˜ d l r c g − (γd + μd ) r d

d d l dt

= − ˜ βd ˜ d l r c g + γd r d + ˜ αd d m

− ˜ βz z d l + γz z d

D g ∂ x g = αg ( d l + φz z ) 4

˜ ϑ g +( d l + φz z ) 4 − βg r g + γg r o g

dr m dt

= φr

(˜ d l

˜ θr + d l − r m

)dd m dt

= φd

(˜ θd

˜ θd + d l − d m

)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n . (18)

Finally crosstalk is modelled via formation of the BZR.DELLA

complex

dz d dt

= βz zd l − γz z d on �n . (19)

Similar to model (7) –(9) , to ensure appropriate units we obtain

rescaled relations between boundary-localised variables in model

(15) - (19) and the corresponding variables in model (10) , (12) –(14) ,

i.e. ˜ r k = l c r k , ˜ r b = l c r b , ˜ z = l c z, ˜ r = l c r, ˜ r o g = l c r o g , ˜ r c g = l c r

c g , ˜ r d = l c r d ,

˜ d l = l c d l , and ˜ z d = l c z d . Since r m

and d m

were already relative quan-

tities with no units there was no need for them to be scaled. To

balance units certain parameters also had to be rescaled: parame-

ters αb , δz , αr , αd , αg , ϑr and ϑd from (10) , (12) - (14) were all mul-

tiplied by l c such that ˜ αb = l c αb etc.; θb , βz and βd were all di-

vided by l c such that ˜ θb = θb /l c etc.; and ϑg was scaled such that˜ ϑ g = l 4 c ϑ g . Since GA molecules have similar size to BR molecules,

we consider D g = D b . All other parameters were taken to be the

same as those used for model (10) , (12) - (14) , i.e. from Tables 1 and

3 and βz = 10 μM

−1 min

−1 , γz = 0 . 133 min

−1 .

System (7), (8), (15) –(19) was solved numerically, and these

solutions were averaged over the space and then compared to

the solutions of system (10) , (12) –(14) . The behaviour of the two

models was similar for most of the parameter sets considered in

Section 6.2 , see e.g. standard cases in Figs. 15 and 17 , however spa-

tially heterogeneous steady states are characteristic for model (7),

(8), (15) –(19) , see Fig. 18 . Spatial heterogeneity does have a signif-

icant effect in the case where perturbations in the phosphoryla-

tion of BZR lead to damped oscillations in the BR signalling path-

way, with much different behaviours of solutions of system (7), (8),

(15) –(19) than for (10) , (12) –(14) , comparing Figs. 15 and 17 . When

the BR signalling pathway is ‘strongly perturbed’, i.e. δz and ρz are

increased sufficiently high, the oscillations in the components of

the BR signalling pathway also cause oscillatory behaviour in the

components of the GA signalling pathway, suggesting that spatial

heterogeneity may contribute to instability of solutions in extreme

cases.

7. Discussion and conclusion

Plants rely on complex integrated hormonal signalling pathways

in order to respond and adapt to changing environmental condi-

tions. The BR and GA signalling pathways have a diverse range

of effects on plant growth and developmental processes, some of

which overlap. To examine the behaviours of these signalling path-

ways and their interactions we developed mathematical models of

the BR signalling pathway, and crosstalk between the BR and GA

pathways under assumptions of both spatial homogeneity: models

(1) and (10) , (12) –(14) , and spatial heterogeneity of signalling pro-

cesses: models (7) –(9) and (7), (8), (15) –(19) .

The parameters in the model for the BR signalling pathway

(2) were determined upon validating the model by comparing its

numerical solutions to experimental data from Tanaka et al. (2005) .

Using numerical optimisation techniques we were able to get a

ood fit for the diverse data sets corresponding to the different

rowth conditions considered in Tanaka et al. (2005) . Our calcu-

ations resulted in values for δz and ρz of a much lower order

f magnitude than the rest of the parameters in the model, sug-

esting that the phosphorylation of BZR occurs on a much slower

ime scale than e.g. perception of BR by BRI1. The model parame-

ers were optimised under two different constraints on the value

f βk , (4) and (5) , corresponding to two different mechanism gov-

rning the interactions of BR, BRI1, and BKI1, where BR and BKI1

inding to and dissociation from BRI1 is instantaneous, or a com-

lex BR.BRI1.BKI1 can be formed, respectively. Most parameters

ad very similar or identical values, with the only notable differ-

nces being for (naturally) βk , and the parameters governing the

hosphorylation of BZR: δz , ρz . Overall our model fitted the data

ell for both mechanisms (4) and (5) , with R 2 values of 0.89 and

.92 respectively, suggesting that both of these mechanisms may

ccurately predict the dynamics of the BR signalling pathway. The

nly growth condition for which the model had a less accurate fit

as for when the growth medium was supplemented with BL.

Qualitative analysis of system (6) revealed that it has only one

teady state for the parameter space P , and that this steady state

s stable in a neighbourhood of the parameter sets defined in

ables 1 and 2 . Bifurcation analysis showed that for a bounded set

f values of ( δz , ρz ), with θz = 41 . 2 μM

−1 and all other parameters

s in Table 1 , system (6) has periodic solutions, i.e. system (6) un-

ergoes a Hopf bifurcation upon varying parameters δz , ρz , corre-

ponding to the dephosphorylation rate and phosphorylation rate

f BZR respectively. No such bifurcations were found when consid-

ring a large range of values for δz , ρz , and θz = 41 . 2 μM

−1 and all

ther parameters as in Table 2 . However, since the values of δz , ρz

or which a bifurcation exists are likely to be biologically unrealis-

ic as they are much greater than the values yielded by validation

f the model by experimental data, this suggests that within some

ange of normal function the BR signalling pathway has good sta-

ility, which is of crucial importance for proper function of plant

issues. Results also suggest that if BR and BKI1 associate and dis-

ociate from BRI1 instantaneously, as in mechanism (4) , stability

f the pathway is principally dependent upon the phosphorylation

tate of BZR. Further if BR.BRI1.BKI1 can be formed, as in mecha-

ism (5) , stability of solutions to model (6) for a wide range of all

arameters ensures effective BR homeostasis.

Numerical solutions for the spatially heterogeneous model (7) –

9) averaged over space and ODE model (2) for the BR signalling

athway demonstrated similar behaviours for the parameter sets

hen the solutions of (7) –(9) and of (2) converge to the steady

tate as t → ∞ . In the oscillatory parameter regime discussed in

ection 4.1 , the PDE-ODE model exhibited distinct behaviour to the

DE model ( Fig. 8 ), suggesting that under such conditions the spa-

ial heterogeneity of the signalling pathway has a large influence

n the dynamics of the molecules involved in the BR signalling

athway.

We investigated the effects of interactions between the BR and

A signalling pathways and the BR-GA signalling crosstalk mech-

nisms ( Li and He, 2013; Tong et al., 2014; Unterholzner et al.,

015 ) by coupling (2), (10) and (11) to obtain model (10) , (12) –

14) , in order to establish which mechanism is more biologically

ignificant ( Ross and Quittenden, 2016; Tong and Chu, 2016; Un-

erholzner et al., 2016 ). We examined the effects of BZR-mediated

iosynthesis of GA and of BZR.DELLA complex formation on the

ehaviour of the BR and GA signalling pathways. We modelled

he cases where there was: no crosstalk, BZR-mediated biosynthe-

is of GA only, BZR.DELLA complex formation only, and both BZR-

ediated biosynthesis of GA and BZR.DELLA complex formation.

he cases of no crosstalk and BZR-mediated biosynthesis exhib-

ted similar behaviour to each other, and the cases of BZR.DELLA

omplex formation and both BZR-mediated biosynthesis of GA and

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 123

Fig. 17. Solutions of system (7), (8), (15) - (19) , averaged over space. For ‘ standard ’ parameter values are as in Tables 1 and 3 and βz = 10 μM

−1 min −1

, γz = 0 . 133 min −1

. For

both ‘ BR perturbed ’ and ‘ BR perturbed, βz / γ z large ’ δz and ρz have had their orders of magnitude increased by two, θz = 41 . 2 μM

−1 , βz / γ z has order of magnitude four, i.e.

βz / γ z ≈ 7500, for ‘ BR perturbed, βz / γ z large ’ only, and all other parameters are as in Tables 1 and 3 . For the case ‘ BR strongly perturbed ’ δz and ρz have had their orders of

magnitude increased by three, θz = 41 . 2 μM

−1 , all other parameters as in Tables 1 and 3 and βz = 10 μM

−1 min −1

, γz = 0 . 133 min −1

.

Fig. 18. The spatial distribution of BR and GA in the cell segment c (solutions of (7), (8), (15) - (19) ) at different times, parameter values are as in Tables 1 and 3 and

βz = 10 μM

−1 min −1

, γz = 0 . 133 min −1

. The concentration of BR starts of uniform, and is produced at x = 7 . 43 μm, the nucleus, leading to the greatest concentration at this

point. The overall concentration increases over the first 8 h, but then decreases steadily, arriving at a spatially heterogeneous steady state after 120 h. The concentration of

GA is also greater at x = 7 . 43 μm where it is produced, but GA attains its spatially heterogenous steady state after 24 h, much more quickly than BR.

B

o

a

c

a

s

w

f

s

fl

b

e

m

s

(

(

s

a

p

c

p

i

a

s

t

ZR.DELLA complex formation exhibited similar behaviour to each

ther ( Fig. 13 ). Further, upon variation of the ratio between DELLA

nd BZR binding thresholds φz the change in behaviour of most

omponents of both pathways was negligible, whereas the vari-

tions in BZR.DELLA binding βz and dissociation γ z rates had a

trong effect upon the BR signalling pathway ( Fig. 14 ). From this

e concluded that BZR.DELLA complex formation has a greater ef-

ect on the behaviour of the BR and GA signalling pathways. This

uggests that interactions between BZR and DELLA exert more in-

uence over the dynamics of the pathways than BZR-mediated

iosynthesis of GA and are more likely to be able to promote any

ffective change in the behaviour of the signalling processes. Nu-

erical results also demonstrated that for all parameter sets where

olutions tended to a steady state, the solutions of model (10) ,

12) –(14) and the averaged over space solutions of model (7), (8),

15) –(19) exhibit similar dynamics, although spatially heterogenous

teady-states are obtained for BR and GA concentrations ( Fig. 18 ).

To examine whether disturbance of one signalling pathway had

greater effect on the other, we modelled the effects of overex-

ression of both BR and GA. Overexpression of BR resulted in large

hanges in the dynamics of the components of the BR signalling

athway only, and overexpression of GA resulted in large changes

n the dynamics of the components of the GA signalling pathway,

nd small changes in the dynamics of the components of the BR

ignalling pathway ( Fig. 16 ). In addition variation of βz and γ z led

o large changes in the dynamics of the BR pathway but only short

124 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

5

w

t

d

s

c

C

t

p

t

f

B

t

c

s

s

μ

w

w

F

d

B

K

w

a

a

t

r

K

[

a

W

s

β

g

s

t

term changes in the dynamics of the GA pathway. From this we

concluded that in general perturbations in the GA signalling path-

way have greater influence on the BR signalling pathway than vice

versa.

To examine the effects of mutations in the BR signalling path-

way on the GA pathway, models (10) , (12) –(14) and (7), (8), (15) –

(19) were solved numerically considering different values for the

parameters δz and ρz governing BZR phosphorylation. For the com-

ponents of the GA signalling pathway solutions to the ODE model

showed different short-term behaviour but tended to the same

steady state, whereas the components of the BR signalling path-

way tended to different steady state. If varying βz and γ z such

that βz / γ z was large in the case when we have damped oscilla-

tion in the components of the BR signalling pathway, the dynam-

ics of some of the components of the GA signalling pathway were

significantly altered but the dynamics of the components of the

BR signalling pathway were unchanged ( Fig. 15 ). For the spatially

heterogeneous model (7), (8), (15) –(19) , in the oscillatory param-

eter regime oscillations in the BR signalling pathway propagated

into the GA signalling pathway ( Fig. 17 ). From this we conclude

that only in the case where there is disturbance in both the BR

signalling pathway and the mechanism of crosstalk between the

BR and GA signalling pathways the dynamics of molecules in the

BR signalling pathway have greater influence on the dynamics of

molecules in the GA signalling pathway. This, together with the

results discussed in the previous paragraph, suggests that under

normal crosstalk the stability of individual signalling pathways is

maintained even if there are disturbances in the other pathway.

To conclude, our analysis of new mathematical models for BR

signalling pathway and the crosstalk between the BR and GA sig-

nalling pathways, derived here, provide a better understanding of

the dynamics of signalling processes, dependent on model param-

eters and spatial heterogeneity. Our results for the BR signalling

pathway highlight its stability, and suggest that this stability is par-

ticularly dependent on the mechanisms governing the phospho-

rylation state of BZR and the subcellular locations of these pro-

cesses. Our results suggest that direct interaction between BZR

and DELLA exerts a larger influence on the dynamics of the BR

and GA signalling pathways than BZR-mediated biosynthesis of GA,

and hence may be the primary mechanism of crosstalk between

the two pathways. Our analysis indicates that during normal plant

function, the GA signalling pathway exerts more influence over the

BR signalling pathway than BR on GA, but mutations in the BR sig-

nalling pathway cause BR signalling to exert some short time in-

fluence over GA pathway and greater influence when coupled with

disturbances in the crosstalk mechanism. Both BR signalling and

crosstalk between BR and GA signalling are important for plant

growth and development. Our modelling and analysis results also

can be used to model the interactions between growth and sig-

nalling processes in order to better understand the influence of the

BR and GA signalling pathways on growth and developmental pro-

cesses in plants.

Acknowledgments

H.R. Allen gratefully acknowledges the support of an EP-

SRC DTA PhD studentship (Grant no. EP/M508019/1), research of

M. Ptashnyk was partially supported by the EPSRC First Grant

EP/K036521/1 .

Appendix A. Validation of model (2) against experimental data

To improve the optimisation process for parameter estimation

we calculated additional constraints for some parameter values us-

ing experimental data.

We first calculated the level of endogenous BL using a value of

.26 ng g −1 (the midpoint of the reported range in mature plants)

hich was obtained from Wang et al. (2014b ). In order to convert

his into an appropriate value in units of μM, we first need the

ensity of plant material. We estimate this by noting that the den-

ity of cellulose is 1.5 kg L −1 ( Gibson, 2012 ), and that cellulose

onstitutes approximately 50% of plant material ( Piotrowski and

arus, 2011 ). For the outstanding amount, we make an assump-

ion that rest of the material is largely accounted for by cyto-

lasm, which we assume to have an approximately equal density

o water, i.e. 1 kg L −1 . From this we get an estimate of 1.25 kg L −1

or the density of plant material. Finally the molecular weight of

L is 4 80.6 86 g mol −1 ( National Center for Biotechnology Informa-

ion, 2017 ). From this data we calculate the endogenous BR con-

entration as

quantity × density

molecular weight =

(5 . 26 × 10

−6 g kg −1

) × (1 . 25 kg L −1

)

4 80 . 6 86 g mol −1

= 0 . 0137 × 10

−6 mol L −1 = 13 . 7 × 10

−3 μM .

We denote this new constant by [ BR ] 0 , and assume it to be the

teady state concentration of BR.

We write μb in terms of αb , θb , δz , Z tot , ρz , θ z and h z by con-

idering the steady state solution of (2) :

b =

αb

[ BR ] 0

(1 +

(θb

Z tot δz [ BKI1] 0 (1+(θz [ BKI1] 0 ) h z ) ρz + δz [ BKI1] 0 (1+(θz [ BKI1] 0 ) h z )

)h b ) ,

here [ BKI 1] 0 denotes the steady state concentration of BKI1,

hich can be calculated using [ BR ] 0 .

We adopt two approaches for the calculation of [ BKI 1] 0 and βk .

irst we consider the BR.BRI1 dissociation constant, its value to be

ependent on the steady state concentrations of BR, BRI1.BKI1, and

R.BRI1 such that

d =

r ∗k b ∗

r ∗b

, (A.1)

here K d denotes the BR.BRI1 dissociation constant, and r ∗k , b ∗,

nd r ∗b

denote the steady state concentrations of BRI1.BKI1, BR,

nd BR.BRI1 respectively. Taking b ∗ = [ BR ] 0 , and rewriting r ∗k , r ∗

b in

erms of k ∗, the steady state concentration of BKI1, (A.1) may be

ewritten as

d (R tot − K tot + k ∗) = (K tot − k ∗)[ BR ] 0 . (A.2)

Thus we may solve for k ∗ to obtain an expression for [ BKI 1] 0 :

BKI1] 0 =

K tot [ BR ] 0 − K d (R tot − K tot )

K d + [ BR ] 0 , (A.3)

nd by taking K d = 11 . 2 nM, the midpoint of values reported in

ang et al. (2001) , calculate that [ BKI1] 0 = 34 . 1 × 10 −3 μM.

Using this value for [ BKI 1] 0 in conjunction with the steady state

olution of the second equation in ( 2 ), we write

k =

(K tot − [ BKI1] 0 )[ BR ] 0 (R tot − K tot + [ BK I1] 0 )[ BK I1] 0

βb

= 0 . 329 βb ,

iving us an expression for βk in terms of βb .

For the second method of calculating [ BKI 1] 0 and βk , we con-

idered that BRI1 has a state where both BR and BKI1 are bound

o it, i.e. the full receptor-based dynamics would look like

db

dt = −β1 br k + γ1 r bk

dk

dt = −β2 kr b + γ2 r bk

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 125

Table A.4

Transformed parameters when fitting the non-dimensionalised model (A.8) against ex-

perimental data from Tanaka et al. (2005) , and using (3) and (4) . Due to the nature of

the non-dimensionalisation, the dimensional parameters θ b and Z tot cannot be found.

Constant Value Units Constant Value Units

βb 8.33 μM

−1 min −1 βk 2.73 μM

−1 min −1

αb 0.27 μM min −1 μb 3.58 min −1

δz 9 . 97 × 10 −4 μM

−1 min −1 ρz 1 . 30 × 10 −4 min −1

θ z 4.05 μM

−1 h z 6

Table A.5

Transformed parameters when fitting the non-dimensionalised model (A.8) against ex-

perimental data from Tanaka et al. (2005) , and using (3) and (5) . Due to the nature of

the non-dimensionalisation, the dimensional parameters θ b and Z tot cannot be found.

Constant Value Units Constant Value Units

βb 8.06 μM

−1 min −1 βk 2 . 11 × 10 −2 μM

−1 min −1

αb 0.27 μM min −1 μb 3.68 min −1

δz 1 . 77 × 10 −3 μM

−1 min −1 ρz 4 . 33 × 10 −4 min −1

θ z 3.98 μM

−1 h z 6

Table A.6

Fitted model parameters obtained by fitting model (2) under conditions (4), (3) to experimental data for each growth con-

dition. Here WT1 corresponds to the case where wild-type plants were grown under control conditions, BZR1 to the case

where wild-type plants were grown in a medium containing BRZ, WT2 to the case where wild-type plants had been grown

in a medium containing BRZ 5μM for two days, BL2 to the case where wild-type plants had been grown in a medium con-

taining BRZ 5μM for two days and then supplemented with 0.1μM BL, MUT3 corresponds to the case where mutant plants

were grown under control conditions, BZR4 to the case where mutant plants were grown in a medium containing BRZ, MUT4

to the case where mutant plants had been grown in a medium containing BRZ 5μM for two days, BL4 to the case where

mutant plants had been grown in a medium containing BRZ 5μM for two days and then supplemented with 0.1μM BL.

WT1 BRZ1 WT2 BL2 MUT3 BRZ3 MUT4 BL4

βb 8.33 8.49 8.50 8.42 2 . 36 × 10 −3 2 . 46 × 10 −3 2 . 43 × 10 −3 2 . 31 × 10 −3

βk 2.73 2.72 2.72 2.80 3.26 ×10 −12 3.10 ×10 −12 2.95 ×10 −12 3.09 ×10 −12

αb 0.27 6.29 ×10 −2 6.29 ×10 −2 6.08 ×10 −2 0.27 0.27 0.28 0.27

θ b 1.96 2.01 2.01 2.11 1.96 1.96 1.92 1.88

μb 3.58 3.54 3.54 13.98 3.50 3.40 3.40 3.57

δz 1.02 ×10 −3 1.05 ×10 −3 1.07 ×10 −3 1.10 ×10 −3 1.02 ×10 −3 1.01 ×10 −3 1.01 ×10 −3 1.06 ×10 −3

Z tot 2.65 2.72 2.72 2.75 2.65 2.58 2.58 2.58

ρz 1.33 ×10 −4 1.39 ×10 −4 1.32 ×10 −4 1.32 ×10 −4 1.34 ×10 −4 1.33 ×10 −4 1.29 ×10 −4 1.22 ×10 −4

θ z 3.95 4.10 4.30 4.06 3.76 3.93 3.91 3.91

h z 6.02 5.82 5.52 5.65 5.95 5.75 5.76 5.76

w

a

r

s

t

r

K

w

β

β

b

a

β

g

t

d d

dr k dt

= −β1 br k + γ1 r bk

dr b dt

= −β2 kr b + γ2 r bk

dr bk

dt = β1 br k + β2 kr b − γ1 r bk − γ2 r bk (A.4)

here the variable r bk denotes the concentration of BR.BRI1.BKI1,

nd β1 and β2 ( γ 1 and γ 2 ) represent the association (dissociation)

ates of BR and BRI1.BKI1, and BKI1 and BR.BRI1 (BR.BRI1.BKI1) re-

pectively. Thus the dissociation constant of BR and BRI1.BKI1, and

he dissociation constant of BKI1 and BR.BRI1, denoted K d and K m

espectively, may be defined by

K d =

γ1

β1

,

m

=

γ2

β2

.

Assuming r bk to be constant allows us to rewrite (A.4) as

db

dt = − γ2 β1

γ1 + γ2

br k +

γ1 β2

γ1 + γ2

kr b ,

dk

dt = − γ1 β2

γ1 + γ2

kr b +

γ2 β1

γ1 + γ2

br k ,

dr k dt

= − γ2 β1

γ1 + γ2

br k +

γ1 β2

γ1 + γ2

kr b ,

dr b dt

= − γ1 β2

γ1 + γ2

kr b +

γ2 β1

γ1 + γ2

br k , (A.5)

hich means that we may not only define βk and βb as

k =

γ1 β2

γ1 + γ2

,

b =

γ2 β1

γ1 + γ2

, (A.6)

ut that we may further divide βk through by βb to show that

K d

K m

=

βk

βb

,

nd hence

k =

K d

K m

βb .

A value of K m

= 4 . 28 μM was reported in Wang et al. (2014a ),

iving an expression βk =

(2 . 62 × 10 −3

)βb . The steady state solu-

ion of the second equation in (2) may thus be written as

( k ∗) 2 +

((R tot − K tot ) +

K m

K

[ BR ] 0

)k ∗ − K m

K

K tot [ BR ] 0 = 0 , (A.7)

126 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

fi

b

z

t

g

w

B

z⎛⎜⎜⎝

c

a

a

w

A

s

d

m

which may be solved for k ∗, and hence an expression for [ BKI 1] 0 is

found

[ BKI1] 0 =

1

2

√ ((R tot − K tot ) +

K m

K d

[ BR ] 0

)2

+ 4

K m

K d

K tot [ BR ] 0

−1

2

((R tot − K tot ) +

K m

K d

[ BR ] 0

),

giving [ BKI1] 0 = 61 . 3 × 10 −3 μM.

In order to check whether a more accurate estimation of the

parameters was given by fitting a non-dimensional model which

does not scale time by one of the fitting variables:

d b

d t = βk (κ − 1 + k ) k − βb (1 − k ) b + μb

(1

1 + ( θb z ) h b − b

)

εd k

d t = βb (1 − k ) b − βk (κ − 1 + k ) k

d z

d t = δz (1 − z ) k − ρz

z

1 + ( θz z ) h z (A.8)

where

βb = 60 βb K tot , βk = 60

βk ( K tot ) 2 μb

αb

, κ =

R tot

K tot ,

μb = 60 μb , θb = θb Z tot , ε =

K tot μb

αb

,

δz = 60 δz K tot , ρz = 60 ρz , θz = θz K tot ,

(A.8) was also fitted to the experimental data. The parameters

whose dimensional values could be found had close agreement

with those found fitting model (2) , Tables A.4 and A.5 , and so it

is better to consider the fitting of the dimensional model (2) to

the experimental data.

Appendix B. Linearised Stability Analysis for the Model (6) of

the BR Signalling Pathway

B1. Proof of positive invariance of M = [0 , 1 + βk κ] × [0 , 1] × [0 , 1] .

To prove that M is positive invariant, we first consider M 1 :={ (b, k, z) ∈ R

3 | b, k, z ≥ 0 } and show the non-negativity of solu-

tions of (6) . Restricting b = 0 , we obtain that f · (1 , 0 , 0) = βk (κ −1 + k ) k +

1

1+(θb z) h b

, and similarly that f · (0 , 1 , 0) =

βb ε b restricting

k = 0 , and f · (0 , 0 , 1) = δz k restricting z = 0 . Since b, k, z ≥ 0 we

also obtain that f (u ) · n (u ) ≥ 0 ∀ u ∈ ∂M 1 and any flow that starts

in M 1 must remain in M 1 for all t , and hence M 1 is a positive in-

variant set.

To show boundedness of solutions of (6) , consider M 2 :={ (b, k, z) ∈ R

3 | b ≤ 1 + βk κ, k ≤ 1 , z ≤ 1 } . Restricting b = 1 + βk κ,

we obtain that f · (−1 , 0 , 0) = (βb (1 − k ) + 1)(1 + βk κ) − βk (κ −1 + k ) k − 1

1+(θb z) h b

, and similarly that f · (0 , −1 , 0) =

βk ε κ restrict-

ing k = 1 , and f · (0 , 0 , −1) =

ρz

1+(θz k ) h z restricting z = 1 . Note that

f · (−1 , 0 , 0) has no critical points for 0 ≤ k ≤ 1, and has value

βb (1 + βk κ) for k = 0 , and 0 for k = 1 . Thus f (u ) · n (u ) ≥ 0 ∀ (u ) ∈∂M 2 and any flow that starts in M 2 must remain in M 2 for all t ,

hence M 2 is a positive invariant set.

Thus M := M 1 ∩ M 2 is a positive invariant region ( Amann, 1990 ).

B2. Calculations for the proof of Theorem 2

Any steady state ( b ∗, k ∗, z ∗) of (6) satisfies

0 =

1

1 + (θb z ∗) h b

− b ∗,

0 = βb (1 − k ∗) b ∗ − βk (κ − 1 + k ∗) k ∗,

0 = −ρz z ∗

1 + (θz k ∗) h z + δz (1 − z ∗) k ∗.

Using simple algebraic manipulation on the system above, we

nd b ∗ and z ∗ in terms of k ∗

∗ =

1

1 +

(

θb

δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

)) h b

=

βk ( κ − 1 + k ∗) k ∗

βb ( 1 − k ∗) ,

∗ =

δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

) .

Then k ∗ is defined as a root of the following non-linear func-

ion

(k ∗) : = βk ( κ − 1 + k ∗) k ∗

⎛

⎝ 1 +

(

θb δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

)) h b

⎞

⎠

− βb ( 1 − k ∗) .

The derivative of g

dg

dk ∗= βb + βk (κ − 1 + 2 k ∗)

⎛

⎝ 1 +

(

θb

δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

)) h b

⎞

⎠

+ βk (κ − 1 + k ∗) k ∗

×

⎛

⎝

h b θb δz ρz

(1 + (1 + h z )(θz k

∗) h z )

(ρz + δz k ∗(1 + (θz k ∗) h z )

)2

(

θb δz k ∗(1 + (θz k

∗) h z )

ρz + δz k ∗(1 + (θz k ∗) h z

)) h b −1

⎞⎠

hich is positive for all p ∈ P and k ∗ ∈ [0, 1].

3. Characteristic equation

The Jacobian for system (6) evaluated at the steady state ( b ∗, k ∗,

∗) is given by

−βb (1 − k ∗) −1 βk (κ − 1 + 2 k ∗) + βb b ∗ − h b θb (θb z

∗) h b −1

( 1+(θb z ∗) h b )

2

βb (1 −k ∗) ε −βb b

∗

ε − βk (κ−1+2 k ∗) ε 0

0 δz (1 − z ∗)+

ρz h z θz z ∗(θz k

∗) h z −1

( 1+(θz k ∗) h z ) 2 −δz k

∗ − ρz

1+(θz k ∗) h z

⎞

⎟ ⎟ ⎠

(B.1)

The coefficients of the characteristic equation associated to Ja-

obian J (B.1) are defined as follows

2 = βb (1 − k ∗) + 1 +

βb

εb ∗ +

βb

ε(κ − 1 + 2 k ∗) + δz k

∗ +

ρz

1 + (θz k ∗) h z ,

1 =

1

ε( βb b

∗ + βk (κ − 1 + 2 k ∗) )

(βb (1 − k ∗) + 2 + δz k

∗ +

ρz

1 + (θz k ∗) h z

)

+ ( βb (1 − k ∗) + 1 )

(δz k

∗ +

ρz

1 + (θz k ∗) h z

),

a 0 =

1

εβb (1 − k ∗)

θb h b (θb z ∗) h b −1

(1 + (θb z ∗) h b ) 2

(δz (1 − z ∗) +

ρz z ∗θz h z (θz k

∗) h z −1

(1 + (θz k ∗) h z ) 2

)

+

1

ε(βb b

∗ + βk (κ − 1 + 2 k ∗))

(δz k

∗ +

ρz

1 + (θz k ∗) h z

),

here ( b ∗, k ∗, z ∗) is a steady state of the system (6) .

ppendix C. Non-dimensionalisation of model (7) - (9) of the BR

ignallng pathway

Since in the PDE-ODE model (7) - (9) some of the variables are

efined on the boundaries we must consider them in units of

ol / m

2 instead of mol / m

3 , compared to the ODE model ( 2 ). To

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 127

a

r

t

θ

l

(

l

w

D

β

δ

A

p

G

o

D

t

D

t

d

s

dapt the units, we scale by the length of the cell segment, i.e.

˜ k = l c r k , ˜ r k = l c r k and ˜ z = l c z. To preserve the balance of units in

he system, we must scale the following parameters ˜ αb = l c αb ,˜ b = θb /l c , ˜ δz = l c δz .

Applying this scaling and non-dimensionalising via t = t ∗t , x = c y, b = b ∗b , k = k ∗k , ˜ r k = r ∗

k r k , ˜ r b = r ∗

b r b , ˜ z = z ∗z , z p = z ∗p z p , system

7) - (9) is transformed to

∂ t b =

D b t ∗

(x ∗) 2 ∂ 2 y b − μb t

∗b

∂ t k =

D k t ∗

(x ∗) 2 ∂ 2 y k

∂ t z p =

D z t ∗

(x ∗) 2 ∂ 2 y z p

⎫ ⎪ ⎬

⎪ ⎭

in c , (C.1)

− D b t ∗

(x ∗) 2 ∂ y b =

βk t ∗r ∗

b k ∗

x ∗b ∗ r b k − βb t ∗r ∗

k

x ∗ b r k

− D k t ∗

(x ∗) 2 ∂ y k =

βb t ∗r ∗

k b ∗

x ∗k ∗ b r k − βk t ∗r ∗

b

x ∗ r b k

d r k d t

= βk k ∗t ∗r b k − βb b

∗t ∗b r k

d r b d t

= βb b ∗t ∗b r k − βk k

∗t ∗r b k

− D z t ∗

(x ∗) 2 ∂ y z p = 0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �c , (C.2)

D b t ∗

(x ∗) 2 ∂ y b =

l c αb t ∗

x ∗b ∗(1+( θb z

∗ z

l c ) h b )

D k t ∗

(x ∗) 2 ∂ y k = 0

d z d t

=

l c δz t ∗ p ∗k ∗z ∗ z p k − ρz t

∗ z

1+(θz k ∗ k ) h z

D z t ∗

(x ∗) 2 ∂ y z p = − l c δz t ∗k ∗

x ∗ z p k +

ρz t ∗z ∗

x ∗ p ∗z

1+(θz k ∗ k ) h z

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n . (C.3)

Take t ∗ = 1 /μb , x ∗ = l c , b ∗ = αb /μb , k ∗ = K tot , r ∗k

= l c R tot , r ∗b

= c R tot , z

∗ = l c Z tot and p ∗ = Z tot . Then

∂ t b = D b ∂ 2 y b − b

∂ t k = D k ∂ 2 y k

∂ t z p = D z ∂ 2 y z p

⎫ ⎪ ⎬

⎪ ⎭

in c . (C.4)

−D b ∂ y b = βk r b k − βb b r k

−D k ∂ y k = ε1

(βb r k b − βk r b k

)d r k d t

= ε2

(βk r b k − βb b r k

)d r b d t

= ε2

(βb b r k − βk k

∗t ∗r b k )

−D z ∂ y z p = 0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �c , (C.5)

D b ∂ y b =

1

1+( θb z ) h b

D k ∂ y k = 0

d z d t

= δz z p k − ρz z

1+( θz k ) h z

D z ∂ y z p = −δz z p k + ρz z

1+( θz k ) h z

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n , (C.6)

here

¯ b =

D b

μb l 2 c

, D k =

D k

μb l 2 c

, D z =

D z

μb l 2 c

, βk =

βk R tot K tot

αb

,

¯b =

βb R tot

μb

, ε1 =

αb

K tot μb

, ε2 =

αb

R tot μb

, θb = θb Z tot ,

¯z =

δz K tot

μb

, ρz =

ρz

μb

, θz = θz K tot .

ppendix D. Rigorous reduction of the model for GA signalling

athway from Middleton et al. (2012)

Here we present the rigourous reduction of the model for the

A signalling pathway from Middleton et al. (2012) to the system

f ODEs (10) and (11) for GA, GID1, DELLA, GA.GID1 o , GA.GID1 c ,

ELLA .GA .GID1 c , GID1 m

and DELLA m

. To do this we assume that

he dynamics of the actual biosynthesis may be captured from the

ELLA dynamics only. We also assume only one functional form of

he DELLA.GID1.GA complex. To save space we denote the indepen-

ent variables by x i for i = 1 , . . . , 21 , in the same order as the full

tatement of the model in Middleton et al. (2012) .

To describe GA signal transduction:

dx 1 dt

= −l a x 1 x 11 + l d x 2 + δgid1 x 20 − μgid1 x 1 , (D.1a)

dx 2 dt

= l a x 1 x 11 − l d x 2 + px 3 − qx 2 , (D.1b)

dx 3 dt

=−px 3 +qx 2 −(u a 1 + u a 2 ) x 6 x 3 +(u d1 + u m

) x 4 +u d2 x 5 , (D.1c)

dx 4 dt

= u a 1 x 6 x 3 − (u d1 + u m

) x 4 , (D.1d)

dx 5 dt

= u a 2 x 6 x 3 − u d2 x 5 , (D.1e)

dx 6 dt

= −(u a 1 + u a 2 ) x 6 x 3 + u d1 x 4 + u d2 x 5 + δdel l a x 21 . (D.1f)

To describe GA biosynthesis:

dx 7 dt

= ω ga 12 − k a 12 x 7 x 16 + k d12 x 12 − μga x 7 , (D.2a)

dx 8 dt

= −k a 15 x 8 x 16 + k d15 x 13 + k m 12 x 12 − μga x 8 , (D.2b)

dx 9 dt

= −k a 24 x 9 x 16 + k d24 x 14 + k m 15 x 13 − μg1 x 9 , (D.2c)

dx 10

dt = −k a 9 x 10 x 17 + k d9 x 15 + k m 24 x 14 − μga x 10 , (D.2d)

dx 11

dt = P mem

S root

R root (A 1 ω ga 4 − B 1 x 11 ) + k m 9 x 15 − l a x 1 x 11 + l d x 2

− μga x 11 . (D.2e)

To describe complexes of GAs and enzymes:

dx 12

dt = k a 12 x 7 x 16 − (k d12 + k m 12 ) x 12 , (D.3a)

dx 13

dt = k a 15 x 8 x 16 − (k d15 + k m 15 ) x 13 , (D.3b)

dx 14

dt = k a 24 x 9 x 16 − (k d24 + k m 24 ) x 14 , (D.3c)

dx 15

dt = k a 9 x 10 x 17 − (k d9 + k m 9 ) x 15 . (D.3d)

To describe the enzymes:

dx 16

dt = −k a 12 x 7 x 16 − k a 15 x 8 x 16 − k a 24 x 9 x 16 + (k d12 + k m 12 ) x 12

+ (k d15 + k m 15 ) x 13 + (k d24 + k m 24 ) x 14 + δga 20 ox x 18

− μga 20 ox x 16 , (D.4a)

128 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

k

w

t

k

w

α

a

θ

T

dt x 6 + θdel l a

dx 17

dt = −k a 9 x 10 x 17 + (k d9 + k m 9 ) x 15 + δga 3 ox x 19 − μga 3 ox x 17 . (D.4b)

To describe the mRNAs:

dx 18

dt = φga 20 ox

(x 6

x 6 + θga 20 ox

− x 18

), (D.5a)

dx 19

dt = φga 3 ox

(x 6

x 6 + θga 3 ox

− x 19

), (D.5b)

dx 20

dt = φgid1

(x 6

x 6 + θgid1

− x 20

), (D.5c)

dx 21

dt = φdel l a

(θdel l a

x 6 + θdel l a

− x 21

). (D.5d)

We set Eq. (D.1e) to zero since we assume that only one con-

figuration of the DELLA.GID1 c .GA 4 can form. Since the conversion

of GA 4 precursors occurs on a relatively fast scale we set the

time-derivatives in (D.2a) - (D.2d) and (D.3a) - (D.5b) equal to zero.

We immediately remove 3 variables by noting from Eqs. (D.1e) ,

(D.5a) and (D.5b) that

x 5 =

u a 2

u d2

x 6 x 3 , (D.6a)

x 18 =

x 6 x 6 + θga 20 ox

, (D.6b)

x 19 =

x 6 x 6 + θga 3 ox

, (D.6c)

and further from Eqs. (D.2a) - (D.2d) , and (D.3a) - (D.4b) that

x 16 =

δga 20 ox

μga 20 ox

x 6 x 6 + θga 20 ox

, (D.7a)

x 17 =

δga 3 ox

μga 3 ox

x 6 x 6 + θga 3 ox

, (D.7b)

k m 9 x 15 = ω ga 12 − μga (x 7 + x 8 + x 9 + x 10 ) , (D.7c)

with

x 7 =

ω ga 12

μga + K 12

, (D.8a)

x 8 =

ω ga 12 K 12

( μga + K 12 ) ( μga + K 15 ) , (D.8b)

x 9 =

ω ga 12 K 12 K 15

( μga + K 12 ) ( μga + K 15 ) ( μga + K 24 ) , (D.8c)

x 10 =

ω ga 12 K 12 K 15 K 24

( μga + K 12 ) ( μga + K 15 ) ( μga + K 24 ) ( μga + K 9 ) , (D.8d)

where

K 12 =

δga 20 ox k a 12 k m 12

μga 20 ox (k d12 + k m 12 )

x 6 x 6 + θga 20 ox

,

K 15 =

δga 20 ox k a 15 k m 15

μga 20 ox (k d15 + k m 15 )

x 6 x 6 + θga 20 ox

,

K 24 =

δga 20 ox k a 24 k m 24

μga 20 ox (k d24 + k m 24 )

x 6 x 6 + θga 20 ox

,

K 9 =

δga 3 ox k a 9 k m 9

μga 3 ox (k d9 + k m 9 )

x 6 x 6 + θga 3 ox

.

Substituting these into Eq. (D.7c)

m 9 x 15 = ω ga 12

1 − μga

⎡

⎢ ⎢ ⎢ ⎣

⎛

⎜ ⎝

(μga + K 15

)(μga + K 24

)(μga + K 9

)+ K 12

(μga + K 24

)(μga + K 9

)+ K 12 K 15

(μga + K 9

)+ K 12 K 15 K 24

⎞

⎟ ⎠

(μga + K 12

)(μga + K 15

)(μga + K 24

)(μga + K 9

)

⎤

⎥ ⎥ ⎥ ⎦

!

= ω ga 12 K 12 K 15 K 24 K 9 (

μga + K 12

)(μga + K 15

)(μga + K 24

)(μga + K 9

) ,

hich can be multiplied through by the Hill function denomina-

ors, and upon simplifying to fourth order terms in x 6 becomes

m 9 x 15 = αg

x 4 6

x 4 6

+ θga

,

here

g =

ω ga 12 δga 3 ox (δga 20 ox ) 3 k a 12 k m 12 k a 15 k m 15 k a 24 k m 24 k a 9 k m 9 ⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(μga μga 20 ox (k d12 + k m 12 ) + δga 20 ox k a 12 k m 12 )

×(μga μga 20 ox (k d15 + k m 15 ) + δga 20 ox k a 15 k m 15 )

×(μga μga 20 ox (k d24 + k m 24 ) + δga 20 ox k a 24 k m 24 )

×(μga μga 3 ox (k d9 + k m 9 ) + δga 3 ox k a 9 k m 9 )

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

nd

ga =

(

μ4 ga θga 3 ox (θga 20 ox )

3 μga 3 ox (μga 20 ox ) 3 (k d12 + k m 12 ) ∗

∗(k d15 + k m 15 )(k d24 + k m 24 )(k d9 + k m 9 )

)

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(μga μga 20 ox (k d12 + k m 12 ) + δga 20 ox k a 12 k m 12 )

×(μga μga 20 ox (k d15 + k m 15 ) + δga 20 ox k a 15 k m 15 )

×(μga μga 20 ox (k d24 + k m 24 ) + δga 20 ox k a 24 k m 24 )

×(μga μga 3 ox (k d9 + k m 9 ) + δga 3 ox k a 9 k m 9 )

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

hus, we obtain a reduced model that reads

dx 1 dt

= −l a x 1 x 11 + l d x 2 + δgid1 x 20 − μgid1 x 1 , (D.9a)

dx 2 dt

= l a x 1 x 11 − l d x 2 + px 3 − qx 2 , (D.9b)

dx 3 dt

= −px 3 + qx 2 − (u a 1 + u a 2 ) x 6 x 3 + (u d1 + u m ) x 4 , (D.9c)

dx 4 dt

= u a 1 x 6 x 3 − (u d1 + u m ) x 4 , (D.9d)

dx 6 dt

= −u a 1 x 6 x 3 + u d1 x 4 + δdel l a x 21 , (D.9e)

dx 11

dt = P mem

S root

R root (A 1 ω ga 4 − B 1 x 11 ) + αg

x 4 6

x 4 6

+ θga

− l a x 1 x 11

+ l d x 2 − μga x 11 , (D.9f)

dx 20

dt = φgid1

(x 6

x 6 + θgid1

− x 20

), (D.9g)

dx 21 = φdel l a

(θdel l a − x 21

). (D.9h)

H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131 129

A

s

s

a

e

s

t

w

c

c

W

t

r

r

a

ε

ε

s

c

m

ppendix E. Non-dimensionalisation of the model including

patial heterogeneity in the crosstalk signalling

Again applying scaling and non-dimensionalising via t = t ∗t etc,

ystem (15) - (19) transforms int o

∂ t b =

D b t ∗

( x ∗) 2 ∂ 2 y b − μb t ∗ b

∂ t k =

D k t ∗

( x ∗) 2 ∂ 2 y k

∂ t z p =

D z t ∗

( x ∗) 2 ∂ 2 y z p

∂ t g =

D g t ∗

( x ∗) 2 ∂ 2 y g − μg t

∗g

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

in c ,

nd receptor based interactions of BR, BKI1 and BRI1 and influx of

xogenous GA occur on the plasma membrane

− D b t ∗

(x ∗) 2 ∂ y b =

βk t ∗r ∗

b k ∗

x ∗b ∗ r b k − βb t ∗r ∗

k

x ∗ r k b

− D k t ∗

(x ∗) 2 ∂ y k =

βb t ∗r ∗

k b ∗

x ∗k ∗ r k b − βk t ∗r ∗

b

x ∗ r b k

− D z t ∗

(x ∗) 2 ∂ y z p = 0

− D g t ∗

(x ∗) 2 ∂ y g = 0

d r k d t

=

βk r ∗b k ∗t ∗

r ∗k

r b k − βb b ∗t ∗r k b

d r b d t

=

βb r ∗k b ∗t ∗

r ∗b

r k b − βk k ∗t ∗r b k

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �c .

We assume that production of BR, change in phosphorylation

tatus of BZR and interactions between BZR and DELLA occur in

he nucleus

D b t ∗

(x ∗) 2 ∂ y b =

l c αb t ∗

1+ (

θb z ∗ z

l c

)h b

D k t ∗

(x ∗) 2 ∂ y k = 0

d z d t

=

l c δz t ∗z ∗p k

∗

z ∗ z p k − ρz t ∗ z

1+(θz k ∗ k ) h z − βz t

∗

l c z d l +

γz z ∗d t ∗

z ∗ z d

D z t ∗

(x ∗) 2 ∂ y z p = − l c δz t ∗k ∗

x ∗ z p k +

ρz z ∗t ∗

x ∗z ∗p z

1+(θz k ∗ k ) h z

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n .

We assume all constituent processes of the GA signalling path-

ay apart from degradation and influx of GA also occur in the nu-

leus

d r d t

= −βg t ∗g ∗r g +

γg r o∗g t

∗

r ∗ r o g +

l c αr t ∗

r ∗ r m

− μr t ∗r

d r o g

d t =

βg r ∗g ∗

r o∗g

r g − (γg + λc ) t ∗r o g +

λo r c∗g t ∗

r o∗g

r c g

d r c g d t

=

λc r o∗g t

∗

r c∗g r o g − λo t ∗r c g − βd d

∗l t ∗

l c d l r

c g +

(γd + μd ) r ∗d t ∗

r c∗g r d

d r d d t

=

βd d ∗l r c∗g t

∗

l c r ∗d d l r

c g − (γd + μd ) t

∗r d

d d l d t

= −βd r c∗g t

∗

l c d l r

c g +

γd r ∗d t ∗

d ∗l

r d +

l c αd t ∗

d ∗l

d m

− βz z ∗t ∗z d l +

γz z ∗d

d ∗l

z d

D g t ∗

( x ∗) 2 ∂ y g =

l c αg t ∗

x ∗g ∗

(d l + φz z

∗d ∗

l z

)4

l 4 c ϑ g

( d ∗l ) 4 +

(d l + φz z ∗

d ∗l

z

)4 − βg r ∗t ∗

x ∗ r g +

γg r o∗g

x ∗g ∗ r o g

d r m d t

= φr t ∗(

d l l c θr d ∗

l + d l

− r m

)d d m d t

= φd t ∗( θd

d ∗l

θd d ∗

l + d l

− d m

)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n .

Finally we assume cross-talk via formation of the BZR.DELLA

omplex also occurs in the nucleus

d z d

d t =

βz z ∗d ∗

l

z ∗d

z d l − γz t ∗z d on �n .

e set

∗ =

1

γz , x ∗ = l c , b ∗ =

αb

μb

, k ∗ = K tot , r ∗k

= l c R tot ,

∗b

= l c R tot , z ∗ = l c Z tot , z ∗p = Z tot , r ∗ =

l c αr

μr , r o∗

g =

l c αr

μr ,

c∗g =

l c αr

μr , r ∗

d =

l c αr

μr , d ∗

l =

l c αd

μd

, g ∗ =

αg

μg , z ∗

d = l c Z tot ,

nd introduce the new dimensionless parameters

D b =

D b

γc l 2 c

, D k =

D k

γZ l 2 c

, D z =

D z

γz l 2 c

, D g =

D g

γz l 2 c

,

μb =

μb

γz , μg =

μg

γz , βk =

βk R tot K tot μb

γz αb

, βb =

βb R tot

γz ,

1 =

αb

μb K tot , ε2 =

αb

μb R tot , θb = θb Z tot , δz =

δz K tot

γz ,

ρz =

ρz

γz , θz = θz K tot , βz =

βz αd

γz μd

, βg =

βg αg

γz μg ,

γg =

γg

γz , μr =

μr

γz , λc =

λc

γz , λo =

λo

γz ,

βd =

βd αd

γz μd

, γd =

γd

γz , μd =

μd

γz , ε3 =

αr μd

μr αd

,

4 =

Z tot μd

αd

, ϑ g =

ϑ g μ4 d

α4 d

, ε5 =

αr μg

μr αg , φr =

φr

γz ,

ϑ r =

ϑ r μd

αd

, φd =

φd

γz , ϑ d =

ϑ d μd

αd

.

Neglecting ¯ s we write out the non-dimensionalised PDE-ODE

ystem, with diffusion and hormone degradation occurring in the

ytoplasm

∂b ∂t

= D b ∂ 2 x b − μb b

∂k ∂t

= D k ∂ 2 x k

∂z p ∂t

= D z ∂ 2 x z p

∂g ∂t

= D g ∂ 2 x g − μg g

⎫ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎭

in c . (E.1)

Receptor binding and dissociation occurring on the plasma

embrane

−D b ∂ x b = βk r b k − βb r k b

−D k ∂ x k = ε1 ( βb r k b − βk r b k )

−D z ∂ x z p = 0

−D g ∂ x g = 0

dr k dt

= ε2 ( βk r b k − βb r k b )

dr b dt

= ε2 ( βb r k b − βk r b k )

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �c . (E.2)

130 H.R. Allen, M. Ptashnyk / Journal of Theoretical Biology 432 (2017) 109–131

v

F

G

G

G

G

H

I

J

K

K

L

L

L

L

M

M

N

P

R

R

S

S

S

S

S

The BR signalling processes occurring in the nucleus

D b ∂ x b =

μb

1+(θb z) h b

D k ∂ x k = 0

dz d t

= δz z p k − ρz z

1+(θz k ) h z − βz zd l + z d

D z ∂ x z p = −δz z p k + ρz z

1+(θz k ) h z

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎭

on �n . (E.3)

The GA signalling processes occurring in the nucleus

dr dt

= −βg rg + γg r o g + μr (r m

− r)

dr o g

dt = βg rg − (γg + λc ) r o g + λo r c g

dr c g dt

= λc r o g − λo r c g − βd d l r c g + (γd + μd ) r d

dr d dt

= βd d l r c g − (γd + μd ) r d

dd l dt

= ε3

(−βd d l r

c g + γd r d

)+ μd d m

− ε4 ( βz zd l + z d )

D g ∂ x g = μg ( d l + ε4 φz z )

4

ϑ g + ( d l + ε4 φz z ) 4 − ε5

(βg rg + γg r

o g

)dr m dt

= φr

(d l

ϑ d + d l − r m

)dd l dt

= φd

(ϑ d

ϑ d + d l − d m

)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

on �n . (E.4)

The crosstalk interactions occurring in the nucleus

dz d dt

= βz zd l − z d on �n . (E.5)

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