Modelling cell division and endoreduplication in tomato fruit pericarp

Mochamad Apri a,b,c,n, Johannes Kromdijk d, Pieter H.B. de Visser d, Maarten de Gee a,b,Jaap Molenaar a,b

a Biometris, Wageningen University and Research Center, 6708 PB Wageningen, The Netherlandsb Netherlands Consortium for Systems Biology, 1090 GE, Amsterdam, The Netherlandsc Industrial and Financial Mathematics Group, Bandung Institute of Technology, Bandung 40132, Indonesiad Greenhouse Horticulture, Wageningen University and Research Center, The Netherlands

A U T H O R - H I G H L I G H T S

� We model cell division in tomato fruit pericarp and its transition to endoreduplication.� The model combines a cell cycle genetic regulatory network and auxin action.� We show that auxin changes can cause transition from cell division to endoreduplication.� Furthermore, the combined action of auxin to the cell cycle regulators improves the robustness of this shift.

a r t i c l e i n f o

Article history:Received 16 September 2013Received in revised form18 January 2014Accepted 23 January 2014Available online 31 January 2014

Keywords:Cell cycleEndoreduplicationMathematical modelTomato fruit pericarpPhytohormone auxin

a b s t r a c t

In many developing plant tissues and organs, differentiating cells switch from the classical cell cycle to analternative partial cycle. This partial cycle bypasses mitosis and allows for multiple rounds of genomeduplication without cell division, giving rise to cells with high ploidy numbers. This partial cycle isreferred to as endoreduplication. Cell division and endoreduplication are important processes forbiomass allocation and yield in tomato. Quantitative trait loci for tomato fruit size or weight arefrequently associated with variations in the pericarp cell number, and due to the tight connectionbetween endoreduplication and cell expansion and the prevalence of polyploidy in storage tissues, afunctional correlation between nuclear ploidy number and cell growth has also been implicated(karyoplasmic ratio theory). In this paper, we assess the applicability of putative mechanisms for theonset of endoreduplication in tomato pericarp cells via development of a mathematical model for the cellcycle gene regulatory network. We focus on targets for regulation of the transition to endoreduplicationby the phytohormone auxin, which is known to play a vital role in the onset of cell expansion anddifferentiation in developing tomato fruit. We show that several putative mechanisms are capable ofinducing the onset of endoreduplication. This redundancy in explanatory mechanisms is explained byanalysing system behaviour as a function of their combined action. Namely, when all these routes toendoreduplication are used in a combined fashion, robustness of the regulation of the transition toendoreduplication is greatly improved.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Tomato (Solanum lycopersicum) is one of the most importantvegetable crops worldwide and the recent publication of the tomatogenomic sequence has enormously increased our knowledge at thegenetic level (Consortium, 2012). To use this improved geneticunderstanding in explaining phenotypic behaviour, functional linksneed to be established. In this paper we aim to improve under-standing of the processes and interactions that underlie the forma-tion of tomato fruits, by looking at phytohormonal influences on cellcycle regulation. Using a modelling approach, we zoom in on a keycontrol point during tomato fruit development: the transition fromthe canonical cell cycle (Fig. 1A) to the partial cycle of endoredupli-cation (Fig. 1B) in tomato pericarp cells.

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

Journal of Theoretical Biology

0022-5193/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jtbi.2014.01.031

Abbreviation: CDK, cyclin dependent kinase; CKI, cyclin dependent kinaseinhibitor; APC/C, anaphase promoting complex; KRP/ICK, Kip-related protein/Interactor of CDKs, family of CKIs; E2F, family of transcription factors; DP,dimerization partner; DEL, dimerization partner – E2F – like protein, alternativename for a-typical E2F factors; CCS52A, cell cycle switch protein 52A; RBR1,rhetinoblastoma-related protein; SIM/SMR, SIAMESE/SIAMESE-related plant-specific family of CKIs; CYC, cyclin, controls the cell cycle progression; SKP, S-phasekinase-associated protein; SCFSKP2A, SKP, Cullin, F-box, SKP2; PROPORZ1, PutativeArabidopsis Transcriptional Adaptor Protein, an Arabidopsis gene, important for theswitch from cell proliferation to differentiation in response to the changes of phyto-auxin and cytokinin concentrations

n Corresponding author.E-mail address: m.apri@math.itb.ac.id (M. Apri).

Journal of Theoretical Biology 349 (2014) 32–43

In most fleshy fruits, growth starts with intense cell division,which after the first weeks gradually declines and is replaced bycell enlargement (Gillaspy et al., 1993). During this expansionphase, individual cells spectacularly increase in volume: more than10,000-fold in tomato mesocarp cells! (Cheniclet et al., 2005). Inmany fleshy fruits, as well as in maize endosperm and Arabidopsistrichomes, this huge cell expansion is accompanied by an increasein ploidy through the process of endoreduplication, i.e., anincomplete cell cycle in which cells continue to replicate theirDNA without subsequent mitosis (Bourdon et al., 2010). Theendoreduplication cycle is a developmental, by default irreversibleprocess, which in tomato pericarp tissue marks the onset ofdifferentiation in parenchyma cells.

Endoreduplication (sometimes also referred to as endocycle orendoreplication) is widespread in nature. It can be observed,e.g., in mammals, drosophila (Sher et al., 2013), yeast (Labib et al.,1995), and higher plants (Cheniclet et al., 2005). Numerous studiesafter the role of endoreduplication in nature have been conducted(Lee et al., 2009). It is found that endoreplication is fundamental forearly development, e.g., in Drosophila melanogaster females endor-eplication is employed to provide nutrients and proteins required tosupport egg production. Endoreduplication is also utilized for tissueregeneration under stress conditions. E.g., the negative effect of waterdeficit on leaf size in Arabidopsis can be reduced by increasing thelevel of endoreduplication. In mammals and higher plants such astomato endoreduplication is employed to enable growth.

In view of the importance of endoreduplication, much researchhas been devoted to find the mechanisms that regulate the transitionbetween canonical cell division and endoreduplication. Here, wefocus on targets for regulation of the transition to endoreduplicationby the phytohormone auxin, which plays a vital role in the onset ofcell expansion and differentiation in developing tomato fruit.

The state of the art of molecular control and functioning ofendoreduplication has recently been reviewed by De Veylder et al.(2011) and more specifically for tomato by Chevalier et al. (2011).In plants, normal G2-M progression is supposed to require significantactivity of the ‘mitosis promoting factor’ (MPF). In plants the MPF iscomposed of the plant M-phase specific cyclin-dependent kinaseCDKB1;1, which is activated by the A-type cyclin CYCA2;3. Thequantitative presence of MPF is a major control factor in determiningwhether the cell divides mitotically or undergoes repeated rounds ofduplicating its DNA without subsequent mitosis (Boudolf et al.,2009). Consequently, mechanisms that reduce the activity of thefunctional complex CDKB1;1/CYCA2;3 should inhibit cytokinesis andcould promote endoreduplication. Contrary to the M-phase specific

CDKB1;1, A-type CDKs (referred to as CDKA) are needed both for G1-S and G2-M transitions. As a consequence, the transition frommitoticto endoreduplicating cycles could also be sensitive to factors influen-cing CDKA activity. In the following paragraphs, we summariseputative mechanisms involved in the onset of endoreduplication.

1.1. Proteolytic degradation of M-phase specific cyclins

To form an active complex, CDKs depend on the presence ofactivating cyclins. Specific degradation of M-phase specific cyclins,such as the A-type cyclin CYCA2;3, could therefore promote endor-eduplication. In the ubiquitin-mediated proteolysis pathway, the E3ubiquitin ligase anaphase promoting complex/cyclosome (APC/C)selectively labels proteins for destruction (for reviews see Capronet al., 2003; Peters, 2006), based on the binding of the APC to theactivating proteins CDH1 or CDC20 (Vodermaier, 2001). It was shownin Boudolf et al. (2009) that CCS52A (the higher plant orthologue ofCDH1) affects the stability of CYCA2;3 in Arabidopsis. A furtheranalysis on the APC activating subunits in tomato in Mathieu-Rivetet al. (2010) showed that SlCCS52A overexpression in young devel-oping fruits led to significant alterations in cell division and DNAploidy levels after eight days post-anthesis (dpa), whereas in fruitsyounger than eight dpa, cyclin transcription rates were suggested tobe high enough to render the cell cycle progression insensitive toCCS52A expression (Joubès and Chevalier, 2000).

CCS52A expression is regulated by E2F transcription factors(Vlieghe et al., 2005; Lammens et al., 2008). In Arabidopsisthaliana, the E2F family of transcription factors is composed ofsix transcription factors E2F A–F and two dimerization partnersDP-A and DP-B (Mariconti et al., 2002). The interplay betweenE2F transcriptional factor with Retinoblastoma-related protein 1(RBR1) forms an important regulator of the expression of manyprominent cell cycle control genes. Typical E2F factors A, B, and Cdimerize with a DP to gain high DNA-binding specificity and canmanipulate transcription via a transactivation domain. In contrast,atypical E2F factors D, E, and F (also called [DP-E2F-LIKE] DEL1-3,see Vandepoele et al., 2002) have two DNA-binding domains, andas a result can bind DNA as monomers. Because E2FD-F/DEL1-3lack the typical transactivation domain, they can inhibit (but notcause) transactivation of E2F responsive elements, by competingfor DNA-binding with E2FA and E2FB (Mariconti et al., 2002). Thea-typical E2FE/DEL1 is directly involved in regulation of CCS52A byrepressing its expression (Vlieghe et al., 2005; Lammens et al.,2008). It was also shown in Berckmans et al. (2011) that E2FB andE2FC have an opposite regulatory effect on E2FE/DEL1, whereas

Fig. 1. Cell cycle in plants. (A) Canonical cell division consists of G1, S, G2, and M phases. In the S phase, the DNA is replicated whereas in the M phase, the nucleus and the celldivide. (B) Endoreduplication cycle. The phases are similar to that in the canonical cell cycle, except that the M phase is bypassed, effectively merging the G2 and G1 phases.Thus, the cell replicates its DNA, but it does not divide.

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–43 33

E2FA (bound to RBR1) has multiple modes of action. The interplayof E2FA and RBR1 directly represses CCS52A and also promotesE2FE/DEL1 expression indirectly, perhaps via the balance of E2FB/E2FC (Magyar et al., 2012).

1.2. CDK inhibition

Alternative to cyclin-specific degradation as a mechanism topromote endoreduplication, CDKs can also be subject to inhibition.By blocking the activity of specific CDKs needed for G2-M transi-tion, CDK inhibitors may also promote the transition from mitoticto endoreduplicating cycles.

1.2.1. KIP-related proteins (KRPs)Kip-related proteins (KRPs), also called Interactors of CDC2-

kinases (ICKs) (Wang et al., 1997), consist of a family of highlyredundant CDK-inhibitors (De Veylder et al., 2001; Vandepoele et al.,2002). Whereas many studies showed KRPs to generally inhibitCDKA;1 activity (e.g. De Veylder et al., 2001; Zhou et al., 2003), somefamily members might also target CDKB (Nakai et al., 2006; Pettkó-Szandtner et al., 2006). An interesting feature of KRPs is the apparentdose-dependency of their inhibitory action. At intermediate concen-trations KRPs inhibit G2-M transition, whereas high concentrationsalso block G1-S (Verkest et al., 2005; Weinl et al., 2005). Consistentwith these results, specific overexpression of SlKRP1 during theexpansion phase in tomato pericarp (when mitotic activity hasalready ceased) led to decreased polyploidy (Nafati et al., 2011).

To explain the KRP dose-dependency, a mechanism was proposedin Verkest et al. (2005), in which the inhibitory target of KRP isCDKA;1. At low levels of inhibition, only mitotic CDKA;1 activity wouldbe sufficiently blocked, whereas more complete inhibition would alsoprevent CDKA;1 activity in endoreduplicating cycles. Rather thanneeding two forms of CDKA;1, it was suggested in De Veylder et al.(2011) that CDKA;1 could simply be needed at lower concentration forendocycle progression, being enough for G1-S transition, but not forG2-M to occur. However, the previously mentioned MPF complex(CDKB1;1/CYCA2;3) and CDKA;1 are interdependent, and the nature ofthis interaction is not completely understood, which complicates theinterpretation of KRP inhibitory action. The interdependency betweenCDKA;1 and CDKB1;1 was proposed in Verkest et al. (2005) to occurvia phosphorylation of KRPs by CDKB1;1. However, antagonisticregulation of expression between CDKB (1 and 2) and CDKA1 intomato pericarp was also shown in Czerednik et al. (2012).

1.2.2. SIAMESE (SIM)/SMR and WEE1 kinaseAlthough in the remainder of the paper we specifically focus on

CDK inhibitory action by KRP, a number of other CDK-inhibitors(CKIs) exist in plants. Two of these CKIs have been suggestedpreviously to be involved in the transition of cell division toendoreplication. Firstly, the SIAMESE (SIM) gene, which was firstidentified in Walker et al. (2000) in Arabidopsis trichomes. SIMand other members of the SIAMESE-related (SMR) family werefound to target CDKA;1 as well as D-type cyclins (Churchmanet al., 2006). As a result, increased expression of an SMR homologwas postulated to be a central factor controlling endoreplicationonset in trichomes, leaves and petals in Arabidopsis (Roeder et al.,2010; Kasili et al., 2010; Roodbarkelari et al., 2010). However, as faras the authors are aware, no expression or action of SIM/SMR hasever been reported for tomato pericarp cells, whereas the CKIactivity of KRP is relatively well-documented. The supposedredundancy between CKI action of SIM/SMR and KRP allowed usto apply the parsimony principle and leave out SIM/SMR of thepresented model analysis.

Secondly, WEE1 kinase mediates the specific inhibition of CDKAvia reversible phosphorylation. This inhibition is proposed to govern

CDK activity during G2 to ensure that DNA replication and DNArepair have finished before mitosis is entered. Subsequent de-phosphorylation occurs at G2/M progression to allow significantCDK activity during M-phase progression (for a review see O'Farrell,2001). WEE1 has been shown to be highly expressed in tissues withvery high nuclear DNA content like maize endosperm (Sun et al.,1999) and tomato pericarp (Gonzalez et al., 2004). Additionally,downregulating WEE1 in tomato, resulted in increased levels ofCDKA and short-cell phenotypes in all examined tissues (Chevalieret al., 2011). These findings are consistent with the putative role ofWEE1 to sustain endocycles in cells with a very high ploidy number(Gonzalez et al., 2007), such as that found in tomato pericarp ormaize endosperm. As there is no evidence that WEE1 is involved inthe transition from full cell cycles to endoreduplication cycles, wedecided to neglect the role of WEE1 kinase in the present analysis,since we are specifically focusing on the transition from full cellcycles to the endoreduplication cycle.

2. Auxin involvement in endoreduplication onset

The regulatory effects of auxin during development of tomatofruit have been known for a long time (Crane, 1964, 1969; Nitsch,1970). To establish an important functional link, in this paper wechoose to use the existing experimental knowledge on involve-ment of auxin in the control of cell cycle regulators (for reviewssee Stals and Inzé, 2001; Richard et al., 2002; Teale et al., 2005) toprovide the rationale behind the selection of candidate parametersin the model analysis of the progression from mitotic to endor-eduplication cycles in tomato pericarp cells. Notice that the mostresults on the effect of auxin on the cell cycle relate to Arabidopsis,not to tomato. However, in the absence of evidence to the contrary,we assume that the same or similar mechanisms apply to tomato,and here we investigate whether these mechanisms may alsosupport the transition to endoreduplication in tomato fruit. If theydo, this will give a new hypothesis on the onset of endoreduplica-tion event in tomato that may be verified by experiments.

2.1. Auxin interactions with E2F

Auxin has many modes of interaction with the expression of E2Ftranscription factors. Because of the E2F network cross-talk, it is hardto discriminate between direct and indirect influences of the auxinto the E2F transcription factor as the promoter sequences of allE2F members, except E2FA, contain E2F-responsive elements. It wasshown by Magyar et al. (2005) that the presence of auxin greatlyenhances E2FB stability, which promotes G1-S and G2-M transitions.E2FC bound to DP-B is involved in the transition from mitotic cellcycles to endoreduplication cycles, by arresting G1-S transition (delPozo et al., 2006). E2FC/DP-B appears to be regulated post-transcriptionally by ubiquitin-mediated degradation (del Pozo et al.,2002), in which the stability of the targeting complex SCFSKP2A isnegatively regulated by auxin (Jurado et al., 2008, 2010).

The presence of atypical E2FD/DEL2 also enhances the expres-sion of several genes involved in cell proliferation. The promoterdomain of E2FD/DEL2 in A. thaliana contains two putative auxinresponse factors and E2FD/DEL2 was shown to be subject tonegative post-transcriptional modification by auxin (Sozzaniet al., 2010). These auxin effects on E2FD could also influenceexpression of other E2F family members as E2FD overexpressionlines appeared to have higher expression of E2FA, E2FB, RBR1, andE2FE/DEL1, which was strongly upregulated (Sozzani et al., 2010).However, compared to wild-type, E2FD mutants did not havealtered levels of E2FE/DEL1.

To summarise, auxin effects via the E2F family and interactionwith RBR1 are integrated in the model via the expression of

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–4334

CCS52A. Interactive effects are difficult to generalize, but byinfluencing the expression of CCS52A, auxin generally promotescell proliferation and represses endocycles.

2.2. Transcriptional regulation of KRP by auxin

The role of PROPORZ1 (PRZ1) in A. thaliana as a mediator ofauxin and cytokinin signals in the control of cell proliferation wasreported by Sieberer et al. (2003). It was shown by Anzola et al.(2010) that PRZ1 in Arabidopsis is needed to modulate histoneacetylation in response to auxin by exposing the effects of PRZ1 ontranscription on the family of KIP related proteins (KRPs), thusproviding a functional link between auxin and KRP expression.PRZ1 appeared to antagonize the repressive auxin signals in theregulation of KRP expression. In the PRZ1-1 mutant, several KRPgenes (as well as E2FC) were misexpressed (Sieberer et al., 2003;Anzola et al., 2010). Overexpression of KRP genes could in partrescue the PRZ1-1 phenotype and silencing of multiple KRP genesled to hyperproliferation.

2.3. Auxin involvement in expression of cyclins or cyclin-dependentkinases

Finally, there is a plenty of circumstantial evidence implicatingauxin involvement in the expression of cyclins or cyclin-dependent kinases. For example, in Ferreira et al. (1994) a severelyreduced expression of CYCA2;1, CYCA2;2, CYCB2;1, and CYCB2;2was shown in suspensions of A. thaliana cells grown in mediumlacking the synthetic auxin α-naphtaleneacetic acid (NAA).CYCB1;1 expression was induced in A. thaliana root cells incubatedin indole-3-acetic acid (IAA) (Ferreira et al., 1994) and incubationin either IAA or NAA also stimulated expression of CDKA;1(Hemerly et al., 1993). Furthermore, reductions in SlCDKB2.1 andSlCyclinB1;1 in transgenic tomato lines with reduced SlARF7(Auxin Response Factor 7) mRNA content were reported (de Jonget al., 2011). However, SlARF7 is also implicated in the regulatorypathway of gibberellic acid, which makes these results difficult to

interpret. It also seems that transcription of several SlCycA genesincreases in leaves of tomato seedlings grown on 10 μM IAA (Guoet al., 2010).

Ishida et al. (2010) showed that depletion of auxin (viainhibition by an auxin antagonist) results in reduction of CYCB1;1and CYCA2;3 expression as well as increased ploidy levels in nucleifrom A. thaliana cotyledons and leaves. However, in the contextof the previously mentioned alternative influences of auxinon endoreduplication onset, the mechanistic nature of this corre-lation remains undecided. However, when auxin preferentiallyaffects mitotic cyclins relative to S-phase cyclins, endoreduplica-tion onset could be promoted by a reduction in auxin.

3. Mathematical model

Due to the importance of endoreduplication, several mathema-tical models have been developed to understand the mechanismsunderlying of the phenomenon in various organisms. In theseorganisms, the dimer of CDK and cyclin appears to be the centralregulator of the cell cycle. For different organisms, different types ofCDKs coordinate the cell cycle progression. For example, fission yeasthas only one CDK, namely CDC2. It is observed that the dimer of CDK/CDC13 is the one that drives the cell cycle progression where itsdynamics triggers the mitosis event. In the absence of this dimer, themodel shows that fission yeast experiences endoreplication (Novakand Tyson, 1997; Sveiczer et al., 2004). Mammalian has four CDKsthat govern the cell cycle. Here, the dimer that controls the G2/Mprogression is cyclin B/CDK1. In this model, it is shown thatendoreplication occurs when the cell lacks of cyclin B/CDK1(Gérard and Goldbeter, 2009). A model is also proposed byRoodbarkelari et al. (2010) to understand endoreduplication inArabidopsis. From this model, they postulate that CCS52A is onlyrequired during the entry phase into endoreduplication.

In this paper, we investigate the mechanisms underlyingendoreduplication in tomato pericarp cells via mathematicalmodel development. The model is obtained by adjusting the cellcycle model from Roodbarkelari et al. (2010) that simulates cell

Fig. 2. A wiring diagram to describe the interactions between the cell cycle regulators. Auxin negatively regulates E2F and PRZ1 productions and positively regulates the CYCA23production. As a result, when auxin level is high, the activities of APC/CCS52A and KRP are repressed relatively strong while CYCA23 is produced relatively high. Hence,endocycle cannot occur.

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–43 35

cycle and endoreduplication in Arabidopsis trichomes. The sub-stantial differences with the previous model are the incorporationof auxin effect in triggering the transition, the incorporation of cellmass, and the elimination of the SIM action in the model. The basisof incorporating auxin effect is explained throughout the paper.The SIM effect is eliminated in our model, as to our knowledge, wecannot find any work that reports the role of SIM in the tomato cellcycle. Therefore, we assume that the inhibition of CDKs is onlyperformed by the KRPs.

Cyclins are produced by ribosomes in the cytoplasm. Oncecyclins have been produced, they move into the nucleus, of whichthe volume is proportional to the number of DNA. Since cyclinproduction increases as the cell grows, the effective activity ofcyclins in the nucleus per DNA also increases. Therefore, weassume that the growth rates of the cyclin concentrations (in thenucleus) are proportional to the ratio of cell mass per DNA (cellmass/DNA), whereas the dynamics of the other protein concentra-tions (in the cell) are not directly influenced by this ratio. Thus, wealso added the influence of cell growth, which was not incorpo-rated in the Arabidopsis model in Roodbarkelari et al. (2010). Theincorporation of cell mass in the model is also done by Chen et al.(2004), Tyson and Novak (2001) and Novak et al. (2001).

The interactions between the cell cycle regulators are shown inFig. 2. As discussed previously, PRZ1 and E2F are negativelyregulated by auxin, whereas CYCA2;3 production is promoted byauxin. Consequently, if the auxin level drops, the CYCA2;3 produc-tion will decrease and hence the amount of MPF reduced. If thereduction is too severe, the cell cannot enter mitosis and endor-eduplication starts to occur. On the other hand, low levels of auxinresult in higher activity of PRZ1 and E2F and, consequently, moreKRP and active APC/CCS52A become available in the cell. Also,trimerization of MPF due to KRP will increase which leads to theinactivation of MPF, and degradation of CYCA2;3 due to that APC/CCS52A will become more pronounced. Eventually, reduction ofMPF occurs and this will trigger endoreduplication.

In addition to MPF, another complex plays an important role inthe progression through the S-phase, which is governed by theA-type CDKs. We refer to this complex as the ‘S-phase PromotingFactor’ (SPF). During endoreduplication, the dynamics of SPF is notsignificantly affected so that the S-phase still occurs.

3.1. Continuous dynamics

As mentioned earlier, cell division in plants is triggered by theactivity of MPF, i.e., the dimer CDKB1;1/CYCA2;3 (Boudolf et al.,2009). In Arabidopsis cells, the transcription factor of CYCA2;3 isactivated through MPF (Csikasz-Nagy et al., 2006; Roodbarkelariet al., 2010). Inspired by the known mechanisms in Arabidopsis,we assume the dynamics of this transcription factor to begoverned by the ODE

d TFA23

dt¼ ðk21pþk21 �MPFÞ � ð1�TFA23Þ

ðJafbþ1�TFA23Þ�k22 �

TFA23

ðJifbþTFA23Þ; ð1Þ

where TFA23 and ð1�TFA23Þ are the fractions of the active andinactive forms of the transcription factor of CYCA2;3, respectively.

The production of cyclin CYCA2;3 is assumed to be influencedby the local auxin level. Additionally, cyclins can be degraded byAPC/C, which is activated by CCS52A (the ortholog of CDH1) andCDC20. Therefore,

d CYCA2;3dt

¼ ðk1pþ f cyca23ðauxÞ � k1 � TFA23Þ �mass�Vdcyca23 � CYCA2;3ð2Þ

where f cyca23ðauxÞ is a function representing the interactionbetween auxin and CYCA2;3 production and Vdcyca23 is given by

Vdcyca23 ¼ k2pþk2pp � APCCCS52Aþk2ppp � APCCDC20; ð3Þ

here APCCCS52A and APCCDC20 denote the fraction of APC/C which isactivated by CCS52A and CDC20, respectively. The molecularmechanisms of APCCCS52A and APCCDC20 are described as

d IEdt

¼ k9 �MPF � ð1� IEÞðJ9þ1� IEÞ�k10 �

IEðJ10þ IEÞ ð4Þ

d APCCDC20

dt¼ k7 � IE � ð1�APCCDC20Þ

ðJ7þ1�APCCDC20Þ�k8 �

APCCDC20

ðJ8þAPCCDC20Þð5Þ

d APCCCS52A

dt¼ ðf E2F ðauxÞ � k3pþk3pp � APCCDC20Þ �

ð1�APCCCS52AÞðJ3þ1�APCCCS52AÞ

�ðk4p � SPFþk4 �MPFÞ � APCCCS52A

ðJ4þAPCCCS52AÞ; ð6Þ

where IE and ð1� IEÞ are the active and inactive forms of ahypothetical intermediary enzyme that is included in the modelto create a delay between the rise of MPF and APC/CCDC20, andf E2F ðauxÞ is a function that models the interaction between auxinand CCS52A via E2F activity.

Once CYCA2;3 is produced, it is assumed to immediately bindto CDKB1;1 to form MPF. Therefore, the dynamics of MPF dependson the dynamic behaviour of CYCA2;3 and the inhibitory activityof KRP. The latter leads to the formation of trimer TrimMPF whichis the inactive form of the MPF. Thus,

d MPFdt

¼ ðk1pþ f cyca23ðauxÞ � k1 � TFA23Þ �massþðlmþVdkrpÞ � TrimMPF

�ðlp � freeKRPþVdcyca23Þ �MPF ð7Þ

d TrimMPF

dt¼ lp � freeKRP �MPF�ðlmþVdcyca23þVdkrpÞ � TrimMPF ; ð8Þ

where

freeKRP ¼ KRP�TrimMPF�TrimSPF ð9Þdenotes the KRP that neither inhibit MPF nor SPF (the proteincomplex that governs progression through G1/S phase).

The dynamics of KRP is described by

d KRPdt

¼ f PRZ1ðauxÞ � k11�Vdkrp � KRP; ð10Þ

Vdkrp¼ k12þk12p � SPFþk12pp �MPF; ð11Þwhere f PRZ1ðauxÞ denotes a function representing the interactionbetween auxin and KRP mediated by PRZ1.

In addition, the dynamics of SPF activity is modelled through

d TFdt

¼ k15p �ð1�TFÞ

ðJ15þ1�TFÞ�ðk16pþk16pp �MPFÞ � TFðJ16þTFÞ ð12Þ

d SPFdt

¼ ðk13pþk13pp � TFÞ �massþðlcmþVdkrpÞ � TrimSPF

�ðlcp � freeKRPþk14pþk14 � APCCDC20Þ � SPF ð13Þ

d TrimSPF

dt¼ lcp � freeKRP � SPF�ðlcmþVdkrpþk14pþk14 � APCCDC20Þ � TrimSPF

ð14Þwhere TF and ð1�TFÞ are the fractions of the active and inactiveforms of the transcription factor of cyclins involved in the forma-tion of the functional SPF complex. TrimSPF denotes the concentra-tion of SPF that is inhibited by KRP. To complete the model, themass/DNA is modelled by the linear ODE

d massdt

¼ μ �mass ð15Þ

with massðt ¼ 0Þ ¼mass0.To simulate the effect of a decrease in the level of auxin, the

functions f cyca23ðauxÞ, f E2F ðauxÞ, and f PRZ1ðauxÞ representing theinteractions between auxin and the cell cycle regulators are

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–4336

described as step functions, with value 1 at the normal (high)auxin level. Thus

f cyca23ðauxÞ ¼1 if auxin is high;α if auxin is low;

(ð16Þ

and similarly for f E2F and fPRZ1.

3.2. Discrete events

DNA is duplicated when the cell has completed the S-phase. Inour model we set this occurrence when the so-called ‘S-phasePromoting Factor’ (SPF) passes a threshold value of 0.25 fromabove. At that moment the cell mass/DNA is halved and subse-quently, the mass/DNA starts building up again according to (15).

A second event is the cell division (cytokinesis) which occurs atthe end of the M phase. In our model, this event is triggered whenthe so-called ‘M-phase Promoting Factor’ (MPF) passes the thresh-old value 0.25 from above. At this moment the cell mass itself ishalved, but the mass/DNA is continuous, and it keeps growingaccording to (15). Thus, we have two discrete events with twocorresponding threshold values; the first is to indicate the DNAduplication which leads to halving of the cell mass/DNA, and thesecond is to indicate the cytokinesis event.

3.3. Parameters of the model

In this model we aim at describing the effect of a decrease inauxin concentration on the onset of endoreduplication through threedifferent mechanisms: reduction of cyclin CYCA2;3 production,increase of APC/CCCS52A activity, and increase of KRP production.Endoreduplication is achieved whenever the dynamics of the MPFremains below the threshold value throughout the cycle. To evaluatethese mechanisms within the presented model structure, suitableparameter values in the model should be used. In our work, these

parameters are found by utilizing parameter sensitivity, which isexplained below. Special attention is paid to parameters k1; k3p, andk11 since they are directly influenced by the concentration of auxin inthe model. To assure endoreduplication, we require

dMPFmax

dk140; ð17Þ

dMPFmax

dk3po0; ð18Þ

dMPFmax

dk11o0: ð19Þ

Thus, the maximum value of MPF decreases when k1 decreases, orwhen k3p or k11 increases.

We started with the parameter values in Roodbarkelari et al.(2010). With this parameter set, the dynamics of the cell division canbe obtained but only (17) could be satisfied. Therefore, the parameterset was adjusted so that the requirements in (18) and (19) could alsobe satisfied. This was done by perturbing the starting parameter setin a direction in which the derivatives in (18) and (19) decrease. Thisdirection was found by calculating the second derivative of (18) and(19) with respect to all parameters. For example, to satisfy (19), wehave to increase parameter ki for which

d2 MPFmax

dk11 dkio0: ð20Þ

Applying this method, we end up with the following parametervalues:

k1 ¼ 0:01, k1p ¼ 0:1, k2p ¼ 0:05, k2pp ¼ 3, k2ppp ¼ 1,k3p ¼ 1, k3pp ¼ 10 k4p ¼ 7:5, k4 ¼ 1:32, J3 ¼ 0:01,J4 ¼ 0:015, k7 ¼ 5:5, k8 ¼ 2:09, J7 ¼ 0:01, J8 ¼ 0:01,k9 ¼ 0:298, k10 ¼ 0:1, J9 ¼ 0:01, J10 ¼ 0:01, k11 ¼ 0:115,k12 ¼ 0:489 k12p ¼ 0:114, k12pp ¼ 0:213, lp¼ 100, lm¼ 1:1,

Fig. 3. The dynamics of the cell cycle. The DNA is duplicated when the SPF passes 0.25 (dashed-line) from above, and hence, the cell mass/DNA is halved. The cell undergoesdivision when the MPF passes 0.25 from above. (A) The overall dynamics, (B) zoom-in of a part (A), containing precisely one cycle of the dynamics, (C) the cell mass/DNAbehaviour. The cell mass and the concentration of the components MPF, SPF and KRP are in arbitrary units (a.u.), APC/CCCS52A is expressed as an activated fraction.

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–43 37

k13p ¼ 0, k13pp ¼ 0:11, k14p ¼ 0:029, k14 ¼ 30, k15p ¼ 0:238,k15pp ¼ 0, k16p ¼ 0:01, k16pp ¼ 2:25, J15 ¼ 0:1, J16 ¼ 0:1,k21p ¼ 0, k21 ¼ 8, k22 ¼ 1:485, Jafb ¼ Jifb ¼ 0:72, lcp¼ 680;lcm¼ 1:56; μ¼ 0:01

4. Results

The mechanistic model, which is discussed in detail above,describes the interactions between the most important cell cycleregulators. It is known that the presence of auxin tends to promotecell division. Thus, when the model is simulated in this condition,we obtain the dynamics of the components shown in Fig. 3. Here,the G1 phase is the interval in which APC/CCCS52A and KRP arevery high whereas MPF and SPF are very low. The APC/CCCS52Aand KRP are then gradually switched off by SPF and MPF so thatSPF and MPF start to rise and the cell enters the S phase. When thecell has completed the S phase, specified when the SPF passes 0.25from above, the DNA is duplicated. Thus, the mass/DNA ratio,which initially increases exponentially, becomes halved when theDNA is duplicated, as shown in Fig. 3C. The cell then enters the G2phase and not long after that the cytokinesis event takes place.This occurs during the M phase which is indicated when the MPFpasses 0.25 from above. At that moment, the cell mass itself ishalved. Thereafter, the cell restarts the cycle from G1 phase again.

In the following paragraphs, we investigate within our modelstructure the feasibility of three different mechanisms for atransition to endoreduplication due to a decrease in local auxinconcentration.

4.1. Auxin interactions with E2F

The effect of auxin interaction with E2F is incorporated inf E2F ðauxÞ in our model. The way we arrive at the parameter valuesfor f E2F ðauxÞ and subsequent parameters at high and low auxin isdescribed in Section 3. When we take

f E2F ðauxÞ ¼1 if auxin is high;2 if auxin is low;

(ð21Þ

we obtain the results shown in Fig. 4.In Fig. 4, initially the auxin level is high. Therefore, the cell

undergoes mitosis whenever MPF passes 0.25 (indicated by dashed-line) from above. After several cell cycles, we simulate a drop in auxinconcentration. This moment is indicated by the arrow in Fig. 4A.As a result, more APC/CCCS52A becomes available in the cell. Thisphenomenon is better visible in the magnification in Fig. 4B, wherethe minimum level of the APC/CCCS52A (red line) is clearly highercompared to the situation before the change in auxin concentration.As a result, the cyclin CYCA2;3 degradation by APC/C becomes morepronounced, even when active APC/CCCS52A is at minimum, whichleads to a decrease in the concentration of MPF.

As can be seen in Fig. 4A, the maximum value of MPF becomeslower than the threshold value, leading to cell cycle arrest.However, the G/S cycle is preserved, which means that the cellstill undergoes endoreduplication. In this process, the cell keepsduplicating its DNA without cell division. Since the cell does notdivide anymore, the cell mass increases exponentially as shown inFig. 4C. However, the cell mass/DNA is still oscillating as shown inFig. 4C.

Fig. 4. The onset of endoreduplication triggered by the interaction between auxin and the E2F transcription factor. In the simulations the decrease of auxin occurs after thesecond G2/M phase is completed, indicated by the arrow in (A). The effect of this decrease on APC/CCCS52A is modelled by increasing f E2F ðauxÞ from 1 to 2. Note that the DNAis duplicated when the S phase is completed, which is triggered when the SPF (green line) passes 0.25 (dashed-line) from above. The cell divides when the MPF (the blueline) passes 0.25 from above. Notice in (B) that, when auxin decreases, the minimum level of APC/CCCS5A (red line) is slightly higher than in the canonical cycle. As a result,in each subsequent cycle the concentration of MPF is decreased so that the maximum value does not reach the threshold value of 0.25 and the mitosis is skipped. Yet, the cellkeeps duplicating the DNA (C). Since it does not undergo mitosis, the cell mass grows exponentially, as can be seen in (D). (For interpretation of the references to colour inthis figure caption, the reader is referred to the web version of this paper.)

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–4338

4.2. Transcriptional regulation of KRP by auxin

The effect of auxin on the KRP expression is represented infunction f PRZ1ðauxÞ in our model

f PRZ1ðauxÞ ¼1 if auxin is high;3 if auxin is low:

�ð22Þ

When these values are plugged into the model, we obtain theresults shown in Fig. 5.

In Fig. 5, initially the auxin level is high. After several cell cycles,we let the auxin concentration decrease, which is indicated by thearrow in Fig. 5A. This leads to an increase in expression of KRP, andincreasing KRP levels in the cell. This condition is described inFig. 5A, where the level of KRP (cyan line) is clearly much higherafter the simulated reduction in auxin concentration. As a result,more MPF is bound to KRP to form trimers and become inactive.Hence, the concentration of the active fraction of MPF decreases.As can be seen in Fig. 5A, the maximum value of active MPFsubsequently becomes lower than the threshold value of 0.25,which blocks mitosis, but DNA duplication still continues in apartial G/S endoreduplication cycle.

4.3. Auxin involvement in expression of cyclins or cyclin-dependentkinases

In our model, we assume that cyclin expression is positivelyregulated by auxin. Therefore, in our model the function f cyca23ðauxÞdecreases when the auxin concentration decreases. When auxinactivity only affects the production of cyclins CYCA2;3, we take

f cyca23ðauxÞ ¼1 if auxin is high;0:3 if auxin is low:

�ð23Þ

When these values are plugged into the model, we obtain the resultsshown in Fig. 6.

We start at a high auxin level so that the cell undergoes mitosis.At the moment indicated by the black arrow in Fig. 6A, the auxinlevel is lowered, and the production of CYCA2,3 decreases. Conse-quently, there is less CYCA2;3 available in the nucleus to bind toCDKB1;1, which causes a decreasing concentration of active MPF.The maximum concentration of MPF eventually becomes lowerthan the threshold value for the mitosis, as shown in Fig. 6A,pushing the cell from the canonical cell cycle into the endoredu-plication mode.

4.4. Combined effects of auxin

In the previous sections we have simulated the independenteffects of auxin on the cell cycle via its interactions withAPC/CCCS52A, KRP, and cyclin CYCA2;3 production. Each mechan-ism on its own can lead to a shift from mitotic cytokinesis toendoreduplication.

The possibility to focus on auxin interaction with each mechan-ism separately is a special feature of the analysis in silico, whereasin vivo, all previously mentioned processes (and many more) acttogether when auxin levels change. We therefore also investigatedthe effect of a decrease in auxin concentration on the cell cycle, viathe combined influences on APC/CCCS52A activity, KRP andCYCA2;3 expression. This was carried out by searching functionvalues for f E2F , fPRZ1, and f cyca23 at low auxin levels that can triggerthe shift to endoreduplication. The result is shown in Fig. 7A.The region of function values that can trigger endoreduplicationare those that lie in between the two surfaces. These surfaces areobtained by discretizing ðf cyca23; f E2F Þ and search for values offPRZ1 that can still produce endoreduplication. Thus, any functionvalue combination that lies within this region will yield

Fig. 5. The onset of endoreduplication triggered by the interaction between auxin and KRP expression. The timing when auxin decreases is indicated by a black arrow. If the effectof the auxin decrease on MPF is modelled by changing f PRZ1ðauxÞ from 1 to 3, the MPF becomes lower than the threshold value (dashed-line), and consequently the cell skipsthe mitosis. Thus, division ceases, but DNA duplication continues (B). Since the cell does not undergo mitosis or cytokinesis, the cell mass grows, as can be seen in (C).(For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–43 39

endoreduplicating cycles. The cross sections of this region throughthe nominal value ðf cyca23; f E2F ; f PRZ1Þ ¼ ð1;1;1Þ which produces thecanonical cell cycle are shown in Fig. 7B.

As can be noticed from Fig. 7, the region that can triggerendoreduplication is quite extensive. This presumably allows formore flexibility in the values of f E2F ðauxÞ; f PRZ1ðauxÞ; and f cyca23ðauxÞ at low auxin level that can trigger the cell to endoreduplicate.

For example, we do not need to increase f E2F ðauxÞ exactly tof E2F ðauxÞ ¼ 2 or to decrease f cyca23ðauxÞ strictly to f cyca23ðauxÞ ¼ 0:3to have endoreduplication. Instead, these values may remaincloser to the nominal values (ðf cyca23; f E2F ; f PRZ1Þ ¼ ð1;1;1Þ). Thisimplies that when the transition to endoreduplication occurs as acombined effect, each individual response requires less sensitivityto declining auxin levels.

Fig. 6. The onset of endoreduplication triggered by the interaction between auxin and CYCA2;3 expression. The timing when auxin decreases is indicated by black arrow. If theeffect of the auxin decrease on CYC2;3 is modelled by changing f cyca23ðauxÞ from 1 to 0.3, the MPF becomes lower than the threshold value (dashed-line), consequently thecell skips the mitosis. Thus, it never divides but keeps duplicating the DNA. Since it does not undergo mitosis, the cell mass grows, as can be seen in (C).

Fig. 7. Region of ðf cyca23; f E2F ; f PRZ1Þ that represents the combination of mechanisms that can trigger endoreduplication. (A) 3D region of the values of ðf cyca23 ; f E2F ; f PRZ1Þ at lowauxin levels. These values represent the auxin interaction with the cell cycle regulators. Any point in between the two surfaces leads to endoreduplication when the auxinlevel in the cell is lowered. Note that the region in this figure is limited to 0:2r f cyca23. At values of f cyca23o0:2 the behaviour of the cycle becomes very irregular, mostprobably because of a numerical artifact due to the small value of f cyca23 � k1p . (B) The cross sections of the 3D region along three axes and through the point where f E2F ¼ 1,f cyca23 ¼ 1, and f PRZ1 ¼ 1. Any point in the grey region leads to endoreduplication when the auxin level in the cell is lowered. The nominal value ðf cyca23 ; f E2F ; f PRZ1Þ ¼ ð1;1;1Þthat produces canonical cell cycle is denoted by ‘*’.

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–4340

5. Discussion and conclusion

In recent years, the knowledge of the genetic level of the onsetand progress of endoreduplication has increased enormously. Wenow realise that this event is the result of interplay between manydifferent cell cycle regulators. As stated in the review on endor-eduplication by De Veylder et al. (2011): “The emerging regulatorycircuitry will probably require computer simulations because thenetwork behaviour will become more and more difficult topredict. A first step has been taken by Roodbarkelari et al. (2010)but now must be extended in the light of the growing under-standing of the mechanism”.

In the present paper, we have developed a mathematical modelto describe the mechanisms that underlie the cell cycle and itstransition to endoreduplication in tomato pericarp. The model isan extension and adjustment of an existing model for Arabidopsis.In contrast to the model from Roodbarkelari et al. (2010), thetriggers that lead to endoreduplication in our model are derivedfrom the regulatory effects of local auxin concentration on cellcycle progression. The effect of auxin is represented in terms ofstep functions that influence the production, the degradation, andthe inhibition of the cell cycle regulators.

5.1. Robust progression from cell division to differentiation

Auxin may trigger endoreduplication by reducing production ofCYCA2;3, increasing activity of APC/CCCS52A, and increasingproduction of KRP. Our results show that each of these mechan-isms on its own may trigger endoreduplication within the pre-sented model structure. Further analysis from the region in Fig. 7shows that the combination of all mechanisms could yield thesame effect. This suggests that the mechanisms that may triggerendoreduplication are not necessarily independent, as wasassumed for the simulations in Figs. 4–6. The existence ofredundancy in the routes to endoreduplication could assure thatthe transition to cell differentiation is more robust. The largeendoreduplication region in the synthesis of the three mechan-isms in Fig. 7 also clearly shows that the combined action isconsiderably more potent in triggering endoreduplication thaneach individual mechanism separately. Presumably this robustnesscould be of considerable importance from a reproductive point ofview. Namely, the transition to endoreduplication and correspond-ing cell differentiation in tomato fruit pericarp ensures timelyexpansion of parenchymal tissues, and is a critical control point inthe development of healthy attractive berries for fruit-feedinganimals to ensure seed dispersal.

5.2. Apparent inconsistency between previous reports on CCS52Aactivity

In Roodbarkelari et al. (2010), a two-step model of endoredu-plication is proposed where CCS52A activity is needed only duringthe entry phase into endoreduplication, and thereafter its leveldecreases to non-significant levels. However, a different result wasshown by Mathieu-Rivet et al. (2010) where considerable CCS52Ais still observed in tomato fruits which, based on their age, areclearly beyond the early cell division phase. From our model, wefound that the need of CCS52A for the progression throughendoreduplication depends on the strength of the interactionbetween auxin and the cell cycle regulators. If the cyclin produc-tion is strongly reduced by auxin depletion while CCS52A activityand KRP production via PRZ1 are only weakly influenced, indeedwe see that the level of APC/CCCS52A activity becomes low, afterthe onset of endoreduplication, as shown in Fig. 6. Otherwise,significant CCS52A activity might remain necessary during theprogression through subsequent cycles of endoreduplication,

which is also clearly observable in the 2D cross-section throughf PRZ1ðauxÞ ¼ 1 in Fig. 7B. Therefore, both results could comply withour simulations. The apparent discrepancy between CCS52A activ-ity in Roodbarkelari et al. (2010) and Mathieu-Rivet et al. (2010)could reflect a difference in the interaction between the cell cycleregulators as mentioned above, between Arabidopsis trichomesand tomato pericarp cells.

5.3. Causal relationship between cell growth and nuclear ploidynumber?

Another adjustment of our model compared to the versionused in Roodbarkelari et al. (2010) is that the effect of cell mass isincorporated in the model. The cell mass increases due to growthand affects the amounts of the cell cycle regulators and resultingcycle events. Based on many experimental observations it turnsout that nuclear ploidy number shows a positive correlation withcell size in polyploid cells like Arabidopsis trichomes (Melaragnoet al., 1993; Hülskamp et al., 1998) or tomato fruit pericarp(Cheniclet et al., 2005). Based on these observations, it has beenhypothesised that polyploidy, due to endoreduplication, couldhave a stimulating influence on cell growth. The functionalcorrelation could follow the ‘karyoplasmic ratio’ theory, accordingto which cells tend to adjust their cytoplasmic volume to thenuclear DNA content (Sugimoto-Shirasu and Roberts, 2003).Although some experimental underpinning for this hypothesishas recently been put forward by Bourdon et al. (2012), such aneffect is not incorporated in our model. However, when we usesimple exponential growth to simulate changes in cell mass, andplot the results against nuclear ploidy number (Fig. 8), it becomesobvious that based on this simplified ploidy versus cell masstrajectory of a single cell, the likelihood of high nuclear ploidynumbers increases linearly with cell size. Although this does notby any means disprove the existence of an influence of polyploidyon cellular expansion, it clearly shows that such a causal relation-ship is not necessary to explain the positive correlation betweencellular size and nuclear ploidy number.

5.4. Conclusions

To conclude, we have developed a model at the cellular level,which was subsequently used to analyse the transition from thecanonical cell cycle to endoreduplication. We evaluated a numberof putative processes that were previously shown to be both

Fig. 8. Modelled cell mass versus ploidy level (C). The nuclear ploidy numberincreases linearly with the cell mass.

M. Apri et al. / Journal of Theoretical Biology 349 (2014) 32–43 41

involved in cell cycle regulation as well as influenced by the localauxin concentration in tomato fruit pericarp cells. As shown in thepresented simulations, these functional links can predict thetransition from cell division to expansion and differentiation as afunction of the prevailing local auxin concentration and theirconcerted action makes the transition more robust.

Although the representation of hormonal regulation by theaction of auxin alone is a gross simplification, as is the representa-tion of the cell cycle regulatory network by a set of only few ODEs,the model can offer a useful analytical tool to mechanisticallyassess previous experimental findings. In addition, the spatiotem-poral distribution of auxin during tomato fruit development andthe function of the PIN gene family have been experimentallyinvestigated by Pattison and Catalá (2012). This gives us possibilityto couple models for auxin diffusion with our cell cycle/endor-eduplication model. In fact this is one of the aims of this researchin the long run. For this we need a growth model that not onlypredicts tomato growth but also incorporates both auxin diffusionvia PINs and our cell cycle model. However, the coupling is still anideal and far beyond the present state of the art.

We also intend this simple model to be a first step in connect-ing different integration levels, which could be integrated as amodule in multi-cellular models for tissue or organ growth(Baldazzi et al., 2012). In such a multi-cellular model it wouldbecome possible to link an emergent decrease in local auxinconcentration, possibly by growth dilution or increasing distanceto auxin sources in the plant, to cell cycle regulation at the level of(groups of) individual cells.

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