Demonstration of competition for flux between neighbouring cells.

<p>In both simulations the bottom left-hand cell is a perfect sink while all other cells generate auxin at a rate of 0<i>∶</i>4 <i>µmolhr⁻</i>¹. (A) Neighbours of the sink share the auxin effluxing from their mutual neighour in the top right-hand corner, so they each receive enough to generate a small concentration of PIN. They also receive identical PIN-mediated flux, even if the top left cell enjoys an initial advantage of 10<i>⁻</i>¹⁰<i>µmol </i><i>µm⁻</i>²<i>;</i> indicating that identical PIN concentrations constitute the stable configuration. (B) Auxin transport between the top two cells is blocked, as indicated by the black zigzag. The top right-hand cell now fluxes auxin exclusively to its lower neighbour. This raises its total auxin efflux enough to express PIN strongly towards the sink, while the sink’s other neighbour has weaker PIN expression than its couterpart in (A). The feedback exponent is given by . (C) This colour bar indicates the auxin concentration in <i>µmol </i><i>µm⁻</i>³ in this figure and the next one.</p

Factors affecting canalization.

<p><b>Summary of main results.</b> The feedback exponent is taken to be greater than one since the linear case never yields canalization. Values other than Y(es) or N(o) indicate necessary conditions.</p

Non-competitive, source-driven model on a randomized tissue grid.

<p>These simulations were run on a tissue generated by randomizing the vertex positions of a square grid. As in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054802#pone-0054802-g005" target="_blank">figure 5</a>, we have zoomed in on the region surrounding the source, for clarity. Both distributions are canalized. They both have <i> = </i>1.5 but the same qualitative result was found for all <i>></i>1, and also for competitive models. (A) In the flux-based simulation the emerging vascular strand twists and turns in response to cell irregularity, while in (B) the flux density-based simulation grows in a straight line, demonstrating that use of the flux-density buffers the PIN dynamics against irregular cell geometries.</p

Cartoon illustration of terms polar-in, polar-out, multi-in and multi-out.

<p>(A) Cell A is multi-in and polar-out. (B) Cell B is polar-in and polar-out. (C) Cell C is polar-in and multi-out. (D) Cell D is multi-in and multi-out.</p

Sink-driven simulation with neighbours removed from the central sink.

<p>While allowing for the sink in these simulations being perfect, they should be compared to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054802#pone-0054802-g009" target="_blank">figure 9</a> B. (A) Neighbours were removed from the central sink. Additional laterals branched from the vein in order to get around the gaps. (B) A vein was cut out when it was five cells long. The two adjacent veins branched in response.</p

Competitive model with <i> = </i>1<i>∶</i>5 on digitized oral meristem.

<p>The cells are repressed for clarity in the left-hand picture. A small region on the surface produces auxin like an L1, with its cells pointing PIN toward the primordium at the centre, while a canalized mid-vein grows into the interior. The right-hand side indicates a rise in auxin is visible at the primordium. Auxin production is limited to the L1 with the value <i>α = </i>0.1 <i>mol </i><i>µm⁻</i>³<i>hr⁻</i>¹. Auxin degradation in the L1 is <i>β = </i>1.0 <i>hr⁻</i>¹, while that in the interior is <i>β = </i>0.4 <i>hr⁻</i>¹. Cell volumes range from about 95 <i>to</i> 300 <i>µm</i>³ with wall areas ranging from six to about sixty <i>µm</i>².</p

Meristem system on a three dimensional tissue consisting of layers of hexagonal prisms.

<p>As required the top layer, representing the L1, has a diffuse PIN allocation pointing towards the primordium (left), from which a straight line of PIN leads down to the sink at the bottom (right). The cells have been repressed for clarity. Auxin production is limited to the L1 with the value <i>α = </i>0.2 <i>mol </i><i>µm⁻</i>³<i>hr⁻</i>¹. Auxin degradation in the L1 is <i>β = </i>1.0 <i>hr⁻</i>¹<i>;</i> while that in the interior is <i>β = </i>0.4 <i>hr⁻</i>¹<i>:</i> Hexagonal faces have an area of 2.60 <i>µm</i>² and the cells’ volume is 2.60 <i>µm</i>³<i>:</i></p

Sink-driven simulations.

<p>In all three cases the sink is an auxin maximum. PIN is ubiquitous and oriented towards the sink (central cell). (A) This simulation has a linear feedback function (<i> = </i>1.0) and the polar-out requirement is not obeyed. (B) This simulation has feedback exponent <i> = </i>1.1. Here the polar-out requirement is obeyed, except where the PIN cut-off is met. (C) This is the same as (B) except that the PIN concentration cap has been halved to 5 <i>µmol </i><i>µm⁻</i>². The build-up of auxin around the sink (centre) is larger in extent and the polar-out requirement is violated more often due to PIN saturation occuring at lower PIN concentrations.</p

Demonstration that available flux alone accounts for PIN generation.

<p>The centre cell is due to be the next cell in a growing vein, represented by the perfect sink at the bottom with the strong PIN expression. The top cell has higher auxin generation than the other two, and the final PIN concentrations are proportional to the square of the auxin generation when <i> = </i>2. Thus conformity to the polar-in requirement is largely unaffected by dynamics within the receiving cell.</p

Exceptional simulation parameters.

<p><b>Exceptional model parameters.</b> Table of parameters and figures in which the parameter values differ from those given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054802#pone-0054802-t002" target="_blank">table 2</a>. An asterisk (*) indicates variation in the relevant variable. An exclamation mark (!) indicates that the qualitative result is relevant over a range of values. In both cases the reader is referred to the figure’s caption.</p