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Cover:The apple is one of the most famous cultural symbols, from the Bible to iPhones. It is also one of the most important

fruit crops in the world. The origin of the apple as we know it today, however, is not entirely clear, and the genetic makeup

of the apples we eat is only just now beginning to be understood. On pages 5765 of this issue of Trends in Genetics,

Amandine Cornille and colleagues discuss genomic data that has illuminated the domestication of the apple and discuss

the genetic history of this common fruit. Cover image from iStock/Sieboldianus.

February 2014 Volume 30, Number 2 pp. 4184

Reviews

Amandine Cornille, Tatiana Giraud,

Marinus J.M. Smulders,

Isabel Roldn-Ruiz, and Pierre Gladieux

Kristin C. Scott and Beth A. Sullivan

Clare Stirzaker, Phillippa C. Taberlay,

Aaron L. Statham, and Susan J. Clark

57 The domestication and evolutionary ecology

of apples

66 Neocentromeres: a place for everything and

everything in its place

75 Mining cancer methylomes: prospects and

challenges

41 Canalization: what the flux?

49 Particle genetics: treating every cell as

unique

Tom Bennett, Genevive Hines, and

Ottoline Leyser

Gal Yvert

Opinions


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Canalization: what the flux?Tom Bennett, Genevie`ve Hines, and Ottoline Leyser

Sainsbury Laboratory, University of Cambridge, Bateman Street, Cambridge, CB2 1LR, UK

Polarized transport of the hormone auxin plays crucial

roles in many processes in plant development. A self-

organizing pattern of auxin transport canalization is

thought to be responsible for vascular patterning and

shoot branching regulation in flowering plants. Mathe-

matical modeling has demonstrated that membrane

localization of PIN-FORMED (PIN)-family auxin efflux

carriers in proportion to net auxin flux can plausibly

explain canalization and possibly other auxin transport

phenomena. Other plausible models have also been

proposed, and there has recently been much interest

in producing a unified model of all auxin transport phe-nomena. However, it is our opinion that lacunae in our

understanding of auxin transport biology are now limit-

ing progress in developing the next generation of mod-

els. Here we examine several key areas where significant

experimental advances are necessary to address both

biological and theoretical aspects of auxin transport,

including the possibility of a unified transport model.

Auxin and self-organization in plant development

The hormone auxin (see Glossary) regulates almost every

aspect of plantdevelopment, and thedirectionalmovement

of auxin by a specialized transport system (polar auxin

transport, PAT) is crucial for many of these processes (Box

1, Figure 1A) [1]. In simple cases, fine-scale redistribution

of auxin allows for differential responses in different cells,

driving patterning and specification events. However, in

many cases patterns are generated not simply by auxin

redistribution but emerge as a property of the system of

feedback between the tissue, auxin, and auxin transport. It

is widely supposed that these developmental systems, and

the auxin transport patterns that drive them, are self-

organizing that is, little or no pre-pattern is needed

[2]. Understanding these apparently self-organising phe-

nomenahas long been an area of interest, as exemplified by

research on phyllotaxis the pattern of leaf initiation at

the shoot meristem (Figure1B) and thevascularpatterns

of leaves (Figure 1C).Because of their self-organizing properties, intuitive

understanding of these systems is difficult and there has

therefore been considerable interest in mathematically

modeling these phenomena [3]. Vascular patterning and

phyllotaxis have primarily been simulated using two fun-

damentally different (but non-exclusive) auxin transport

heuristics, often respectively referred to as with-the-flux

(WTF) and up-the-gradient (UTG) (Box 2). Although these

models have been immensely useful in demonstrating the

plausibility of self-organizing transport as a developmen-

tal mechanism, neither type of model is explicit about their

biological basis, and they include parameters that are not

based in current mechanistic understanding, such as as-

sessment of auxin concentration in neighboring cells. Fur-

thermore, it is probable that neither heuristic is inherently

capable of capturing the full range of self-organizing auxintransport [3]. To understand better the role of self-orga-

nizing auxin transport in plant development, a new gener-

ation of models that are more deeply rooted in a

mechanistic understanding of auxin biology is needed.

However, our understanding of the biology of canalization

and related phenomena has been somewhat outstripped by

theoretical work on these problems, and now represents a

limiting factor for modeling. The purpose of this article is

thusnot topropose anext-generationmodel but to examine

the areas in which we need to improve our understanding

of auxin transport and discuss how current models can be

used to prioritize these experiments. We primarily discuss

WTF

models,

particularly

in

the

context

of

the

canalizationhypothesis, vascular patterning, and shoot branching.

There has recently been considerable interest in attempt-

ing to unify models of auxin transport, and we also assess

prospects for achieving this goal.

The canalization hypothesis of vascular patterning

Vascular patterning in plants is complex but orderly [4]

it is not hardwired but clearly proceeds according to firm

principles such that the same general vascular topology is

reproduced in almost every individual in a species

(Figure 1C). Local auxin application induces vascular dif-

ferentiation in plant tissue, but in narrow strands running

away from the application site, rather than in wide fields of

cells [5]. These observations led to the singular and pio-

neering contributions of Tsvi Sachs, whose elegant experi-

ments are still central to the field [4,68]. Sachs proposed

that as auxin flows through tissues it upregulates and

polarizes its own transport,which gradually becomes chan-

neled or canalized into files of cells with very high

auxin flux away from auxin sources (Figure 1D); these cell

files can then differentiate to form vasculature (Figure 2)

[7,8]. Sachs also demonstrated that new vasculature usu-

ally develops towards and unites with existingvasculature

strands, leading to a connectedvascularnetwork (Figure 2)

[4,7,8]. However, he also demonstrated that existing vas-

culature could be hyper-canalized by the addition of

Opinion

0168-9525/$ see front matter

2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tig.2013.11.001

Corresponding author: Leyser, O. ([email protected]).

Keywords: auxin; auxin transport; self-organization; canalization; mathematical

modeling.

Trends in Genetics, February 2014, Vol. 30, No. 2 41
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auxin, in which case developingvasculature could not find

and unite with it (Figure 2) [7].

The work of Sachs pre-dated the advent of molecular

genetics, and he therefore needed to infer upstream events

based largely on terminal vascular differentiation patterns.

Remarkably, recent investigations have supported his hy-

potheses at a molecular level including the central canali-

zation concept that, from an initially broad domain of cells

with low auxin flux, a subset of cells become progressively

morepolarizedandcompetent to transport auxinandhave

shown that canalization is an important component ofvas-

cular patterning [911]. It should be emphasized that,

although some

auxin

flows

do

undoubtedly

canalize, notall auxin transport phenomena involve canalization. For

instance, initiation of leaf primordia in angiosperm shoot

meristems(Figure 1C) requiresformation ofan auxin maxi-

mum by a focused pattern of transport (maximization)

(Figure 1D) [12]. Canalization has generally been explored

throughWTFmodels (Box 2) which canaccurately simulate

patterns of auxin transport in a number of developmental

processes, includingvascular formation in stems and leaves

[1315]. Canalization of auxin transport has also been

recently modeled as an explanation for the inhibition of

bud outgrowth by actively growing shoots, a scenario in

which thedevelopment ofvasculature isnot directly consid-

ered, although it is an important additional outcome of the

bud activation process [16]. Auxin transport canalizationthus has the potential to explain multiple developmental

phenomena in plants.

What is the flux?

All current models of canalization are based on a large

corpus of research into polar auxin transport, and in

particular the behavior of PIN-family auxin efflux carriers

(Box 1). Examination of phyllotaxis andvein formation has

shownvery distinctive patterns of PIN protein localization

consistent with canalization and maximization [9,12,17].

Almost all modern models of auxin transport therefore

explicitly simulate membrane-localized PIN proteins that

directly

influence

the

amount

and

direction

of

auxin

trans-port. The main difference between the WTF and UTG

models, based on the experimental observations of PIN

protein localization in different scenarios, relates to the

rules for allocating PIN proteins to membranes (Box 2). In

WTF models PIN proteins are allocated to each membrane

in a cell inproportion to flux, thenetquantity of auxin that

exits the cell across that membrane. Net flux efficiently

couples cells together (because high net flux from cell i!jtends to prevent high flux from j!i), allowing cells tocouple to larger-scale patterns of flux and speeding the

emergence of global WTF patterns in the overall direction

i!j (Box 2). Although mathematically this is a very neatsolution, as a concept it is likely to be unrealistic because it

requires a cell to calculate the net exchange of auxin across

its membranes (including passive uptake). There is no

known biological mechanism that achieves this, which is

a common criticism of flux-based models [18]. Neverthe-

less, it is clear that cells in real systems do canalize auxin

transport, and do so by allocating PIN proteins apparently

in proportion to net auxin flux. It is thus the absolute crux

of canalization research to establish how cells are able to

localize PIN proteins in relation to larger-scale patterns in

a self-organizing manner.

The most plausible explanation for the apparent ability

of cells to calculate net flux is that cells measure one or

more other variables, the combined effect of which is

Glossary

Angiosperms: floweringplants.By farthe largest major groupingof plants and

also the most recently evolved. Includes almost all crop species and model

species such as Arabidopsis thaliana.

Apoplast: the space between plant cells, occupied by thick cellulosic walls

(Figure 1A). There is a significant pH difference between the apoplast (pH 5.5)

and cytoplasm (pH 7), and this directly affects auxin transport in accordance

with the chemiosmotic hypothesis.

Arabidopsis thaliana: a principal plantmodel species,particularly formolecular

genetic studies, due to its small size, small genome, andshort life-cycle. Its smallsize, however, means that it is not ideally suited to canalization research.

Auxin, auxin transport: auxin (indole-3-acetic acid, IAA) is a low molecular

weight, long-distance signal with many functions in plant development.

Specific, polar auxin transport (PAT) through tissues seems to be an ancient

characteristic of land plants.

Canalization: an apparently self-organizing pattern of auxin transport in which

an initially broaddomain ofauxin-transporting cells is reduced to a narrow canal.

This is thought to occur by auxin upregulating and polarizing its own transport.

Charophyte algae: a group of green algae that constitute the sister taxon of

land plants.

Chemiosmotic hypothesis: see Box 1.

Gymnosperms: a diverse group of plants, including conifers, that produce

seeds butnot flowers. Togetherwith angiosperms theymake up the seed-plant

(spermatophyte) clade.

Lycophytes: an ancient group of vascular plants; sister taxon to the clade

containing ferns and seed plants.

Maximization: an apparently self-organizing pattern of auxin transport in

which auxin is transported towards cells containing higher concentrations ofauxin, leading to the formation of an auxin maximum.

Meristem: a specialized region of cell division in plants. Shoot meristems in

angiospermsandgymnosperms combinecelldivisionwiththe productionofnew

organs, either leaves or reproductive structures. Shootmeristems in otherplants

are generally simpler in structure and contain far fewer cells. Rootmeristemsare

only present in vascular plants and do not directly produce new lateral organs.

Phyllotaxis: an apparently self-organizing developmental pattern describing

the position of organs (e.g., leaves) along and around the stem. Different

phyllotactic patterns occur in different species. Phyllotaxis in angiosperms

results primarily from the positioning of neworgan primordia on the flanks of

the multicellular shoot meristem, and is established by maximization-like

patterns of auxin transport in the meristem.

PIN auxin efflux carriers (PINs): a family of proteins that are general ly

accepted to be auxin efflux carriers. Canonical PIN proteins have plasma

membrane localizations, often polarized, and are thought to be the principal

determinants of the direction of auxin efflux, in line with the chemiosmotic

hypothesis. Named after a founding member, PIN-FORMED1 (PIN1), in turn

named for its mutant phenotype involving impaired organ initiation at theshoot meristem a result of aberrantmaximization.

PINOID-family kinases: a small family of serine/threonine kinases that

phosphorylate the intracellular loop of canonical PIN proteins, thereby

controlling their localization. Named after the founding member, PINOID, in

turn named for the resemblance of its mutant phenotype to pin1.

Super-linear: a mathematical relationship in which one variable is influenced

by another with a greater than linear effect; examples include quadratic

(y = a x 2), cubic (y = ax3), and exponential (y = ax) functions.

Up-the-gradient (UTG): a modeling heuristic widely used to simulate

maximization-like patterns of auxin transport (Box 2), in which PIN proteins

are allocated to the plasma membrane in proportion to the concentration of

auxin in cells neighboring that membrane.

Vascular patterning: an apparently self-organizing developmental phenomen-

on in which the position of future veins is established by canalization-like

patterns of polar auxin transport through a tissue.

Vascular plants: the plant clade containing angiosperms,gymnosperms, ferns,

and lycophytes. Defined by the presence of a differentiated vascular network.

Non-vascularplantssuchasmosses lackspecialized tissues forwater transportand are limited in their size as a result.

Vasculature/veins: the vascular network in plants plays analogous roles to the

vascular system in animals. I t consists of two paral le l systems, xylem

(primarily water-conducting) and phloem (primarily sugar-conducting), that

generally develop in association with each other.

With-the-flux (WTF): a modelingheuristic widely usedto simulate canalization-

like patterns of auxin transport (Box 2) in which PIN proteins are allocated to

the plasma membrane in proportion to the net flux of auxin through that

membrane.

Opinion Trends in Genetics February 2014, Vol. 30, No. 2

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proportional to net flux. It is not even necessary for these

measurements to include any component of flux, but an

attractive hypothesis is that cells canmeasure transport-

er-mediated efflux of auxin across a givenmembrane, and

combine this with other information to regulate PIN

protein allocation. For instance, it is possible that, as

PIN proteins transport molecules of auxin, they (or a

protein partner) produce a positive-feedback signal that

reduces the removal of those PIN proteins from the

membrane. This alone would be sufficient to maintain

WTF patterns of PIN protein localization, but not to

generate them in the first place, because this mechanism

would not specifically orient PINs on the membrane

opposite an auxin source. To achieve this aspect of PIN

localization presumably requires at least some informa-

tion from outside the cell. It is therefore likely that the

canalization mechanism has at least two components,

and these might include measurement of auxin concen-

trations on either (or both) sides of cell membranes, as for

instance proposed in a recent model of auxin transport in

which extracellular auxin concentration is the major

determinant of PIN allocation [19]. A recently proposed

framework for cell coupling,unrelated to concentration or

flux-based models, but operating through bidirectional

exchange of information across the apoplast, would also

theoretically be able to generate large-scale patterns of

PIN localization [20].

The first step towards testing these ideas must be to

probe the genetic basis of the canalization feedback mech-

anism, using the well-established toolkit inArabidopsis, a

goal distinct from understanding how PIN proteins are

polarized in general [10] or providing descriptive analyses

of the

process

of

canalization

[9,10,21].

A

pure

canaliza-tion system must be established in Arabidopsis, compara-

ble to the original experiments of Sachs even though its

diminutive size makes this difficult but with the addition

of reporter lines such that the early stages of canalization

can be visualized. By using this system to test the canali-

zation response in mutants or under pharmacological

treatments that impair known auxin-sensing, auxin trans-

port, and PIN polarity-generating mechanisms, the role of

those factors in the canalization process can be examined,

helping to narrow down the mechanisms involved. Of

course, as yet undiscovered factors might be central to

the canalization mechanism, in which case screening for

canalization-deficient mutants may be a sensible ap-

proach. The vascular patterning defects seen in pin1pin6 doublemutants [11] provide a possible reference point

for screening for developmental phenotypes, but another

approach to screening may be to look for mutants in which

initially well-established but broad transport domains

(visualized by reporter genes) fail to narrow down, the

hallmark of canalization. Distinguishing potential canali-

zation mutants from generalized auxin transport mutants

will be important, and a sensitized genetic background

might be preferable to help pick out otherwise relatively

subtle phenotypes.

This top-down approach to canalization should be ac-

companied by general research to allow improved param-

eterization

of

auxin

transport

models.

Trying

to

quantify,for example, the amounts of auxin and PIN protein in

different parts of each cell, or the cycling rates for PIN

proteins, will be fiendishly difficult, but even establishing a

loose range would be an improvement over the current

absence of data. Other important questions to address

include whether the relationship between flux (or equiva-

lent) and PIN allocation is linear or super-linear, whether

there is a saturation point for flux-correlated PIN-alloca-

tion, and whether the pool of PIN proteins is quasi-infinite

and freely allocated or limited and proportionately (re-

)allocated according to flux, all aspects that current models

have shown to be potentially important in pattern emer-

gence [3,18,22]. Cell culture-based systems may prove

useful in addressing these questions, as they have been

in dissecting the action of auxin transporters and mecha-

nisms of auxin homeostasis [23,24].

Increased understanding of PIN protein behavior will

aid modeling of auxin transport,

The experiments described above would be well-comple-

mented by bottom-up approaches to improve understand-

ing of the behavior of PIN proteins. There is a significant

body of work relating to the localization of PIN proteins,

and it is known that in some cell types they can cycle

rapidly between the plasma membrane and endosomal

compartments [25]. It is this system that presumably

Box 1. Auxin transport

Auxin is transported in a polar manner through many tissues, and

the canalization theory of Sachs [7,8] is framed in the context of

PAT. Long-distance PAT has often been theorized as connecting

auxin sources (regions of highauxin concentration or production)

to sinks (regions of lowauxin concentration or high turnover) [6].

In most canalizing systems, developing tissue (leaves, buds, etc.)

acts as an auxin source and established vasculature acts as a sink

(Figure 2). More recent work

has shown that vasculature generallyhas high auxin concentrations [45], and therefore sink strength in

this system is probably determined by auxin flux rapidly carrying

auxin away from the source. Subsequent to Sachs initial canaliza-

tion work, a mechanistic basis for PAT was proposed in the

chemiosmotic hypothesis. Central to this is the weakly-acidic

nature of auxin (pKa 4.75), which means that a significant fraction

of auxin molecules in the apoplast (pH 5.5) are protonated and

neutrally charged, and can passively enter cells through the lipid

membrane; however, the largely deprotonated auxin in the

cytoplasm (pH 7) cannot passively exit cells (Figure 1A). Specific

efflux carriers are therefore required to mobilize auxin from cells,

and it wasproposed that polar localization of these proteinswould

explain theoverall polarityof auxin transport [46,47]. Thediscovery

of PIN-family auxin efflux carriers, transmembrane proteins which

often have polar localization [44,48], confirmed the validity of the

chemiosmotic

hypothesis, and it

is generally

accepted that PINproteins are the major determinants of the directionality of local

auxin flux (Figure 1A) [28]. Members of the large ABC family of

auxin transporters seem to act as non-polar auxin efflux carriers

[49], andthere arealsoauxin influxcarriers of theAUX1/LAXfamily

[50] (Figure 1A). Auxin regulates its own transport, andin particular

PIN protein abundance and localization, at multiple levels, both

transcriptional and post-transcriptional [1,51]. For instance, intra-

cellular auxin levels can regulate transcription of PIN genes

through canonical auxin signaling [52], whereas apoplastic auxin

can inhibit PIN endocytosis though the ABP1 receptor [53]. Work in

thi s area has been greatly a facil itated by l ive imaging of

transporters fused to fluorescent proteins [54], and by proxy live-

imaging of intracellularauxin based on fluorescent reporters of the

activity of various components of the transcriptional auxin signal-

ing pathway [55,56].


43
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allows for the dynamic changes in PIN protein localization

necessary for both WTF and UTG patterns to emerge.

However, the mechanisms that determine how PIN pro-

teins are allocated to different membranes in different

situations are poorly understood, despite observations of

the resultant patterns. PIN protein localization can be

influenced by regulatory proteins such as PINOID-family

protein kinases,which phosphorylate the long intracellular

(A) (B)

(C)

(D)

I1

P1

P8

P3

P6

P1

P9P4P7

P2

P10

P5 I1

I2

P11

IAA

IAA

IAA

Cytoplasmp

H7

ApoplastpH5.5

TRENDS in Genetics

Figure 1. Auxin transport, plant development, and self-organization. (A) Schematic illustrating the chemiosmotic mechanism of polar auxin transport. Protonated auxin

(indole-3-acetic acid, IAA) inthe cell wall space theapoplast (green) canmovepassivelyinto cells throughplasmamembranes(black arrows). Influxmayalsobe assisted

by influxcarriers (yellow circles). Deprotonated auxin(IAA) inthecytoplasm canonlymove out ofcells bytheactionof efflux carriers (redcircles), andpolarlocalization of these

carriers (such as PINproteins) generatesoverall polarity in auxin transport. (B) Schematic showing the phyllotactic pattern of organ initiation atArabidopsisshootmeristems

(top-down view).New organsareproduced in a stable spiral pattern, withapproximately 1378 separating eachnew organ. I1 and I2mark theposition of the next twoinitiating

organ primordia to form. P1 (youngest)P11 (oldest) are existing organ primordia. Phyllotaxis is an apparently self-organizing developmental process that involves auxin

maximization, and has primarily been modeled using an up-the-gradient (UTG) heuristic. (C) Schematic showing vascular patterning in an Arabidopsis leaf. The midvein

(purple) forms first and joins the leaf to the main vascular bundles in the stem. First-order veins (dark blue) directly connect to themidvein andareassociatedwithlocal auxin

maximaat theedgeof theleaf. Themajorauxinmaxima associatedwith lobes/serrations areshown in red, others areomitted forclarity. Lower-order veins (light blue) connect

first-orderveinstogetherto form ahighlyconnective reticulatenetworkthatveryefficientlyservesthewholeorgan.Thevascularnetwork is specified byauxintransportthrough

the leafblade, towards themidvein andultimately the stem. Vascular patterning is an apparently self-organizing developmental process that involves auxin canalization, and

has primarily been modeled using a with-the-flux (WTF) heuristic. (D) Schematic cross-section through an Arabidopsisshoot meristem showing organ initiation events at I1

andP1. Auxin in themeristem is transported (blue arrows) towards thesite of I1 by PINproteins (greenbars),resultingin the formationof an auxinmaximum (redshading). At

P1 thepatternof auxin transport is partially reversed,withauxinbeing transported away from themaximum in a down-the-gradient pattern. Only a thin fileof cells transports

auxin,thus showing a canalizedpattern of transport;thesecellswill become themidveinof theneworgan.Organinitiation thus involvesauxin canalizationandmaximizationin

tight spatiotemporal cooperation. Neither WTF nor UTGmodels of auxin transport have yet convincingly captured this complete range of behavior.


44


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loop domain of particular PIN proteins [26,27]. This loop

domain shows extensive variation in structure between

different types of PIN protein (Bennettet al., unpublished),

meaning that each type of PIN protein could have an

inherently different potential for localization; for instance,

disruption of specific loop domains can result in different

localizations within the same cell type [28]. Ultimately,

canalization-like patterns are mediated through specific

regulation of a subset of PIN proteins. A deep structure

function analysis of PIN proteins would therefore delineate

howeachpartof theloopcontributes toPIN localization, and

how each PIN protein behaves under different circum-

stances. Indeed, it is possible that part of the loop in some

PIN proteins is a specific regulatory element for flux-based

feedback, mediating canalization-like behavior of the pro-

tein in effect, a canalization motif. In Arabidopsis PIN1

plays major roles inbothcanalizationand maximization, but

in other species including grasses these two processes may

bemediatedby structurally-distinctPINproteins(OConnor

et al., unpublished). Investigating the possible evolution of

PIN protein structures specialized for canalization may

therefore also provide an entry point for dissecting how

canalization is regulated at a molecular level. Cell type-

specific factors [29], external stimuli including light [30]

and long-distance signals such as cytokinins and strigolac-

tones [31,32] can all influence the localization of PIN pro-

teins, and it is thereforealso important tocontinueassessing

how different combinations of these proximal factors might

contribute in large-scale self-organizing PIN behavior.

The role of the apoplast in auxin transport

A frequent simplifying assumption in modeling auxin

transport is to ignore the apoplast and assume that auxin

is transported directly from one cell to the next. Given the

chemiosmotic basis of auxin transport (Box 1) thismay be a

dangerous omission because apoplastic conditions are

interconnected with auxin transport in multiple ways

[33].For instance, low extracellular pH results in increased

passive movement of auxin into cells (Figure 1A), and

auxin ion export through PIN proteins is likely to be

energized by the proton motive force across the plasma

membrane. Furthermore, a long-established activity of

Box 2. Auxin transport models

Most mathematical models so far published have generally taken a

major experimental observation regarding auxin transport (e.g., PIN

localization towards an auxin maximum), and abstracted it into a

singlemathematical concept (basicmodeling terminology is summar-

ized in Figure I). These observation-based models can be broadly

allocated to two classes flux-based or concentration-based

depending on the primary source of information they use to allocate

PIN proteins to plasma-membranes. In practice all flux-based modelsare explicitly of a WTF subset, and almost all concentration-based

models are UTG. Within each broad class, the exact set of parameters

and level of abstraction varies between models. These models are not

mutually exclusive (mathematically they could be combined), but so

far have been considered separately. A small number of mechan-

istically more explicit models have been proposed, for example one

that proposes that auxin concentration in neighboring cells is

measured via its effects on cell wall stress(alsobased on experimental

observation) [57], although purely theoretical models, for example

based on apoplastic transcriptional auxin gradients, have also been

proposed [19].

WTF PIN allocation

In WTF models a positive feedback loop increases PIN insertion rate

(or decreases PIN removal) in a given cell membrane when there is

increased flux f the net quantity of auxin exported through thatmembrane, per unit time and per unit area.

Mathematically, a general formulation for the dynamics of PIN

concentration ( p) in the membrane section of a cell i facing

neighboring cell j (ij) includes PIN insertion, both at a basal rate (r0)

and at an increased rate given by the auxin flux feedback [f(fij)] and

PIN removal (m).

d pi j

dt

f fij r0 mpi j; x>0r0 mpi j; x 0

The exact feedback relationship between flux and PIN allocation has

important ramifications for model function. Several different and

purely theoretical relationships have been explored in models,

including a simple linear relationship, f(f)= af [40], quadratic,

f(f)= af2 [15], or a Hil l function ff afn/Kfn[16].

UTG PIN allocation

In UTG models, PIN insertion rate is increased in membrane sectionsaccording to the auxin concentrations in cells neighboring those

membranes. PIN proteins in cell i are preferentially inserted in the

section of membrane that faces the neighboring cell jwith the highest

auxin concentration, at the expense of other membranes. This

increases the auxin concentration in j, thus driving positive feedback

of PIN allocation.

Although some models [39,41] explicitly include PIN cycling

between an intracellular pool of non-allocated PINs ( pi) and mem-

brane-bound PINs (pij), the more streamlined model proposed in [38]

assumes that all PINs (pi) are instantly and competitively allocated

between the different membrane sections proportionally to concen-

tration. The sets of equations below describe the two situations; inboth cases, ai is the auxin concentration in cell i and aj is the auxin

concentration in cell jwhich is adjacent to cell ialong the membrane

section ij. The set of cells kadjacent to cell iis the set of neighbors N(i).

d pi j

dt aaj; pi pi v pi j

d pidt

gai; pi m piX

k 2 Ni

aaj; pi pi v pij

8>>>:

39;41

pij aajP

k 2 Niaakpi

d pidt

gai; pi mpi

8>>>:

38

Cell i Cell j

ajai

pij

0

ij

TRENDS in Genetics

Figure I. Basic modeling terminology. Schematic illustrating some of the major

terms used in mathematical modeling of auxin transport, and their interrelation.

The cell i faces its neighborjat themembrane ij. The basal concentration of PIN

protein (pij) in the membrane ij (indicated by a green bar) is determined by the

relative rates of insertion (r0) from an intracellular pool (indicated by a greencircle) and recycl ing from the membrane to the intracellular pool (m). PIN

allocation to the membrane can be increased by positive feedback relating the

either fluxthroughfij (indicated by a blue arrow)or theauxin concentration in cell

j (aj).


45
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intracellular auxin is to stimulate proton-pumping

ATPases, thereby further acidifying the cell wall [34]. This

gives rise to a potential positive feedback loop in which

increased intracellular auxin in one cell, acting through

apoplastic acidification, drives increased auxin uptake in

neighboring cells and increased activity of its in situ PIN

proteins. This mechanism can therefore contribute to the

generation

of

net

flux

between

cells,

particularly

at

highauxin concentrations, and might have important ramifica-

tions in the switching behavior seen during organ initia-

tion (Figure 1D) where auxin accumulation in the

epidermis is associated with internalization and canaliza-

tion of auxin flow. The apoplast is the central focus of a

recently proposed model [19] which invokes a polarity-

generating mechanism that is neither WTF nor UTG,

but instead relies on gradients of auxin across the cell wall

partitioning an extracellular receptor to generate PIN

polarization in the adjoining cells.The apoplast is certainly

a potential source of information for polarization mecha-

nisms but there is little biological evidence to support this

model, which requires steep gradients of auxin in the tiny

apoplastic space to make it work [3,19]. It will certainly be

interesting to test the effect of apoplastic auxin and pH

dynamics in both WTF and UTG models. However, al-

though there are now a range of approaches for assessing

intracellular auxin concentrations (albeit indirectly), there

is currently no way to quantify apoplastic auxin, and tools

to do so should be a priority for the field.

Unification of auxin transport models

The integration of modeling and molecular genetics has

demonstrated that auxin transport dynamics provide a

plausible explanation for vascular patterning and shoot

branching regulation via canalization [9,15,16,18] and for

phyllotaxis via maximization [35,36]. Subsequently, there

has been considerable interest in producing a unifying

model of auxin transport that is capable of reproducing

both canalization and maximization patterns with a single

heuristic and set of parameters. Published models of this

type have mostly been extensions of previous models with

either purely WTF mechanisms [37] or purely UTG mech-

anisms

[38], but

cannot

straightforwardly

reproduce

bothbehaviors because they require significantly altering

parameters in different parts of the simulation, making

biologically improbable assumptions, or ignoring wet lab

data [3]. It is fair to say that the consensus in the field,

supported by reanalysis of current models [3], is that no

satisfactory unifying model has been developed yet per-

haps not surprisingly given the current gaps in our under-

standing. From a biological perspective, an interesting

question is not whether the models can be mathematically

unified after all, with enough parameters one could

model anything [39,40] but whether they should be

unified. Are canalization and maximization really flip-

sides of the same coin or are they fundamentally different

processes using different mechanisms in different tissues?

There are also other auxin transport patterns, particularly

in the root, that do not resemble either canalization or

maximization are all these phenomena essentially the

same process or are they divergent mechanisms that share

only some basic aspects?

This question is particularly intriguing from an evolu-

tionary perspective because PAT is present throughout

land plants and in at least some charophyte algae [41].

However, there is currently little evidence for the specific

phenomena of canalization or maximization outside

angiosperms. Given its importance in vascular develop-

ment, it seems a reasonable hypothesis that canalization

(A) (B) (C) (D) (E)

TRENDS in Genetics

Figure2 . Canalization phenomena. Schematics based on the classic experiments of Sachs on excised pea epicotyls (juvenile stems). Green cylinders indicate naive non-

vascular tissue; gray cylinders indicate vascular bundles. Red semicircles indicate addition of exogenous auxin. Blue lines indicate newly induced vascular strands. (A)

Simple demonstration of canalization: lateral auxin application induces vascular connection with the main vascular bundle. (B,C) Sourcesink relationships in induced

vascular strands. Thevascularbundle is surgically removed and two sources of auxin areadded to the apicalend of theepicotyl. If added simultaneously(B) twonew sets

of vasculature are formed. In both cases canalization occurs towards the site of the former vascular bundle, indicating that it is still a strong sink for auxin. If one auxin

sourceis added subsequent to theother (C), canalization now occursfrom that sourcetowards thenew vascular tissueformed by thefirstsource, indicating that it is now a

stronger sink. (D) Sink-finding in canalization. A cut in the epicotyl does not prevent canalization occurring between an exogenous auxin source and the existing vascular

bundle. (E) Hyper-canalization. Addition of a strong auxin source to the existing vascular bundle now prevents sink-finding by an exogenous auxin source. However,

canalizationand vascular formation fromthe auxin source can stilloccur in a non-connective fashion.Dotted bluelines indicate the discontinuation of the vascularstrands.


46
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evolved early in the vascular plant clade and is present

throughout it. PAT is present in the lycophyte Selaginella

kraussiana and plays a role in vascular development, but

whether this is canalization-driven is currently unclear

[42]. Lycophytes and ferns have meristems with single

apical cells, and initiate organs in a fundamentally differ-

ent manner to seed plants. This suggests that auxin max-

imization

in

meristems

arose

specifically

in

the

largemeristems of the seed-plant lineage although this does

not necessarily preclude maximization-type phenomena in

ferns and lycophytes. If generalized PAT, canalization, and

maximization did evolve at different points in the evolu-

tionary history of plants then sequential innovations could

have generated novel auxin transport phenomena. In turn,

this would suggest that these phenomena are not equal or

equivalent, but require process-specific genetic compo-

nents, which could have included changes in the structure

of the PIN proteins themselves; however, more work is

necessary to establish the exact evolutionary history of

auxin transport. In angiosperms, different PINs are prob-

ably specialized for, or act preferentially in, particular

processes; for instance, the primary (but not sole) functionof PIN2 in Arabidopsis is to control a specific shootward

auxin flux in the root meristem [43,44]. Further investiga-

tion of auxin transport phenomena and PIN protein sub-

functionalization outside angiosperms will not only be

illuminating with regard to the evolution of development

in land plants but will also help in dissecting the nature of

auxin transport itself.

Even though canalization and maximization both in-

volve PIN1 inArabidopsis, there is some molecular genetic

evidence to suggest that they might not be identical pro-

cesses; for instance, PINOID plays a crucial role in maxi-

mization but is less central to canalization [29]. This has

led

to

suggestions

that

there

is

tissue-

or

context-specificswitching between modes of auxin transport, an approach

used in another model [17] in which maximization and

canalization are effectively modeled separately. However,

as with so much in biology, it is likely that the reality will

be more nuanced, especially because we do not yet under-

stand the mechanisms of either canalization or maximiza-

tion. It is plausible that there is a core machinery for

allocating PIN proteins to membranes that, given the in-

herent differences between contexts, is capable of generat-

ing both canalization and maximization and possibly all

auxin transport phenomena. For example, if PIN allocation

is achieved by the combined assessment of two or more

factors inside and outside cells (as discussed above), then

perhaps both patterns can be generated depending on the

weightings given to those different factors in different con-

texts.This coremachinery couldhave beenelaboratedupon

during plant evolution to generate new patterns of auxin

transport, butremainthesame fundamental unifiedmech-

anism.Ultimately, althoughcomputational work can tellus

that themodelsareunifiable,wewillonlyfindoutforsureby

pushing forward our biological understanding of auxin

transport across the whole plant kingdom.

Concluding remarks

The impressive progress of theoretical research into auxin

transport phenomena has outstripped advances in our

biological understanding of these processes, particularly

in the case of canalization, which has only received limited

experimental attention in the recent molecular genetic era

ofplantdevelopment [911,21].Further experiments along

the lines proposed here are now required to gain a deeper

understanding of the canalization mechanism, and must

aim to unite physiological and genetic approaches in a

single

species.

These

will

not

only

be

relevant

to

canaliza-tion itself but also to the auxin transport field more gener-

ally, allowing construction of a new generation of models to

examine self-organizing plant development.

Acknowledgments

Our research is funded by the Gatsby Foundation and the European

Research Council (Project 294514 EnCoDe). We would like to thank

Graeme Mitchison for critical reading of the manuscript.

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