Morphogen-regulated growth and division can violate the ULSR.

<p>(A) Snapshot of a simulation (simulation time 84 h) from an auxin-based developmental model (<i>Model 10</i>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>) in which cell division and (slow) growth are only possible above a fixed threshold of auxin concentration. Below this threshold (13.5 AU) and above a second lower threshold (8.8 AU) cells undergo accelerated growth. From an early stage of growth on strain rates are unbalanced leading to tissue distortion. The malformations accumulate and cell divisions are predominantly taking place in the central layers as determined by the auxin gradient. Colouring according to areal strain rates (‘AS’, <i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>) (B) Snapshot of a simulation with <i>Model 10</i>, but with a more dominant diffusion regime (parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g006" target="_blank">Figure 6D</a>, with as threshold for accelerated growth <60000 AU and for growth termination <40000 AU) leading to a less pronounced lateral gradient (at 90 h). This produces less severe tissue distortion, but still severely inhibits growth and leads to unrealistic cell size distributions. Colouring is according to growth potential (‘GP’, as defined in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a> section) as a measure for ‘turgor pressure’, showing a central region at the apex which opposes growth of the outer cell layers (indicated by blue versus red colours).</p

Putting Theory to the Test: Which Regulatory Mechanisms Can Drive Realistic Growth of a Root?

<div><p>In recent years there has been a strong development of computational approaches to mechanistically understand organ growth regulation in plants. In this study, simulation methods were used to explore which regulatory mechanisms can lead to realistic output at the cell and whole organ scale and which other possibilities must be discarded as they result in cellular patterns and kinematic characteristics that are not consistent with experimental observations for the <i>Arabidopsis thaliana</i> primary root. To aid in this analysis, a ‘Uniform Longitudinal Strain Rule’ (ULSR) was formulated as a necessary condition for stable, unidirectional, symplastic growth. Our simulations indicate that symplastic structures are robust to differences in longitudinal strain rates along the growth axis only if these differences are small and short-lived. Whereas simple cell-autonomous regulatory rules based on counters and timers can produce stable growth, it was found that steady developmental zones and smooth transitions in cell lengths are not feasible. By introducing spatial cues into growth regulation, those inadequacies could be avoided and experimental data could be faithfully reproduced. Nevertheless, a root growth model based on previous polar auxin-transport mechanisms violates the proposed ULSR due to the presence of lateral gradients. Models with layer-specific regulation or layer-driven growth offer potential solutions. Alternatively, a model representing the known cross-talk between auxin, as the cell proliferation promoting factor, and cytokinin, as the cell differentiation promoting factor, predicts the effect of hormone-perturbations on meristem size. By down-regulating PIN-mediated transport through the transcription factor SHY2, cytokinin effectively flattens the lateral auxin gradient, at the basal boundary of the division zone, (thereby imposing the ULSR) to signal the exit of proliferation and start of elongation. This model exploration underlines the value of generating virtual root growth kinematics to dissect and understand the mechanisms controlling this biological system.</p></div

Compact model overview.

<p>*Model 9 differs from Model 8 in that auxin was added as a model variable to study its dynamics in a realistic, expanding cellular grid.</p><p>Overview of the models used in this study. Various categories w.r.t. developmental decisions are presented. Column (3) specifies the transition between division and elongation zone (DZ and EZ, respectively); column (4) specifies the transition to mature (differentiated) cells (mature zone or MZ) based on timing since the release from the QC or a spatial signal at a fixed distance from the root apex; column (5) specifies whether division rate is determined via a timer or sizer mechanism; and column (6) how cellular growth rates are defined. Developmental events can be determined to happen after a fixed duration (‘Timer’), a fixed number of divisions (‘Counter’), a fixed cell size (‘Sizer’), and a fixed distance from the root apex (‘Ruler’). For Models 10–12 more complicated regulatory mechanisms are specified. In Model 4 extra random noise was added to the timer (‘+ noise’). For Model 5 cell division is dependent on a timer mechanism, with a different cell cycle time for inner and outer layers (‘Timer (variant)’). Model 6 uses a uniform size (area) criterion for cell division, irrespectively of cell geometry (‘Sizer (uniform)’). For Model 7 the size criterion is adapted to the difference in the width between inner and outer layers, thereby acting effectively as a length criterion. In the other (non-cell autonomous) cases the cell division sizer differs between layers. It is usually assumed that DZ and EZ have a characteristic elongation rate. More details of the models can be found in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s016" target="_blank">Text S1</a> and in the corresponding figures.</p><p>Compact model overview.</p

Timers and counters can produce stable directional growth.

<p>Example simulations of counter and timer based models for root growth (<i>Model 2</i> and <i>Model 3</i> in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>). Upon release from the QC, cells can divide for a pre-programmed time duration (A–C) or number of times (D), followed by a fixed time duration of accelerated growth. (A) Total root length versus simulation time. It takes approximately 75 hours for the first cells to start accelerating growth (indicated by ‘*’). Subsequent sets of quasi synchronously dividing cells then grow exponentially for a fixed time, yielding a roughly linear area increase. (B) Total cell number versus simulation time. The cell number is built up exponentially till 12 (columns)×2³ = 96 ‘clones’ are formed. These then undergo accelerated growth. One cell cycle later the next pool of 96 cells is ready to do the same and cell numbers increase by constant steps. (C) Detailed snapshots (at 99 hours, with diamond shaped markers, and 108 hours, with circular markers) of approximate cell length along the main growth axis. Subpopulations with narrow length distributions can be seen corresponding to dividing, accelerating and mature cells. The accelerating cell population occupies an increasing area until adding to the mature zone. The ‘polyloc method’ was used for curve fitting (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>). (D) Simulation output of <i>Model 2</i> at 99 h (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>). The imposed growth and division rules have resulted in a highly regular grid with distinct zones of similar cell length (division zone (DZ) and elongation zone (EZ) are indicated). Areal strain rates (‘AS’ as defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>) are mapped on the cellular grid, showing the elongation zone as a distinct region of relatively uniform accelerated growth.</p

Importance of radially uniform (longitudinal) strain rates.

<p>(A) Schematic representation of the ‘ULSR’. Growth of the root apex is simplified as occurring strictly vertically. Arrows indicate that local strain rates are the same at each position along the vertical axis. In our modelling framework one horizontal axis can intersect cells with various average positions, sizes and strain rates thereby inevitably challenging the ULSR. (B–C) Simulation output of <i>Model 1</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>) with growth defined as a constant increment to a cell's target area. Cells in inner files are narrower and therefore longitudinal strain will be higher. This leads to growth irregularity with faster growth for central files creating an imbalance in the observed strain rates and distortion of the originally horizontal cell walls across the diverse layers. (B) Starting state of cell grid. (C) Cell grid depicted at simulation time 108 h.</p

Cytokinin-auxin cross-talk in root development.

<p>Simulation output of <i>Model 12</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>). (A) Schematic view of regulatory interactions between model variables (in <i>italics</i>) and PIN exporters. Dotted lines illustrate potential cross-talk with gibberellin (GA) signalling (auxin stimulating GA and GA inhibiting cytokinin signalling) not included in the model. GA is represented in the model as an independent signal that undergoes growth-dilution, thereby determining the exit from elongation <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910-Band2" target="_blank">[19]</a>. (B) Simulation output at 30 h with blue colouring relative to the SHY2 concentration. A domain with strong SHY2 expression is present. (C–E) Colouring of the cell grid is according to the auxin concentration in arbitrary units (‘AU’). Notice a transition from a (basal) linear gradient to a (apical) 2D gradient dominated by polar transport. This is caused by the PIN inhibition at the SHY2 expression domain. The extent of the division zone (DZ) is indicated. (D) Simulation of this model with a 4-fold stronger auxin source shows that the DZ is expanded. (E) Simulation of this model with a 4-fold stronger cytokinin source shows that the DZ has shrunk considerably. This corresponds to observations from Beemster and Baskin <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910-Beemster3" target="_blank">[34]</a> on treatment of <i>Arabidopsis</i> root with auxin and cytokinin analogues.</p

Smooth developmental transitions through spatial signalling.

<p>Cells are instructed by spatial signals at a fixed distance from the growing root apex (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a> – <i>Model 8</i>). They do not behave as clonal subpopulations and smooth developmental processes are a natural result. (A) Plot of root length versus simulation time shows a smooth transition to a steady linear organ growth (indicated by ‘*’). This is similar to experimental studies (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g001" target="_blank">Figure 1</a> in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910-Beemster3" target="_blank">[34]</a>). (B) Plot of total cell number versus simulation time shows a roughly similar trend as in (A) (‘*’ indicating approximately steady increase). (C) Cell length along the principal growth axis (at simulation time 50 h) demonstrates that the exit of division and start of accelerated growth at a fixed position from the apex can lead to a smooth cell length profile as seen in experimental studies (compare <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g002" target="_blank">Figure 2</a>). Grey circles represent data points across all cell layers, whereas empty circles are data from the 2 outer (here called epidermal) cell layers only. The ‘epidermal’ data points lie roughly within the expected twofold range at each position along the longitudinal axis. The ‘polyloc’ method was used for curve fitting (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>). (D) Simulation output with areal strain rates (‘AS’ as defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>) mapped on the cellular grid, showing the elongation zone with accelerated growth. This represents a snapshot at 45 h from a model similar to <i>Model 8</i> (except a relative growth rate of 0.2 per simulation step of 30 min, occurring between 240–750 µm from the root apex and division at a size of 360 and 180 µm² for outer and inner cell layers, respectively).</p

Auxin dynamics on a dynamic cellular grid.

<p>Output of model simulations (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a> – <i>Model 9</i>) with cells growing exponentially (in accordance with <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910-Ivanov1" target="_blank">[32]</a>) at the same low specific growth rate (except the ‘cap’ cells) up to a fixed distance of 240 µm from the tip and then undergoing 10-fold growth acceleration (up to a specific distance of 750 µm). Cell division was based on a sizer mechanism with noise added to have a more irregular, natural, tissue shape (up to a fixed distance of 240 µm from the tip). To not bias the analysis, growth and division did not depend on auxin. Four simulations based on differences in kinetic parameter values as well as some subtle differences in the differential equations were set up to represent different types of auxin sources in the model (see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s016" target="_blank">Text S1</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s006" target="_blank">Figure S5</a>). (A–C) Auxin profiles over time in the growing root. (A) A strictly external source of auxin (import from upper vascular and border cell walls, export via epidermal cell walls). Kinetic parameters (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s016" target="_blank">Text S1</a>):  = 1 µm;  = 6000 µm².min⁻¹;  = 1200 µm.min⁻¹ = 600 µm.min⁻¹;  = 0.0003 min⁻¹;  = 0 (µm² min)⁻¹;  = 2.10⁷ min⁻¹. The sharp peak at the apex shifts and slowly dilutes out by steady growth. (B) A strictly internal auxin source (production rates proportional to cell areas). Kinetic parameters that are different from (A):  = 100 (µm² min)⁻¹;  = 0 min⁻¹. The shape of the curve is as in (A) without peak dilution. (C) A strictly internal auxin source (production rates constant on a per cell basis). Kinetic parameters that are different from (A):  = 10⁴ min⁻¹;  = 0 min⁻¹. A similarly shaped curve is shown as (A) and (B), but with peak convergence as time progresses. (D) Steady auxin concentration pattern (yellow colouring according to arbitrary concentration units ‘AU’) on a dynamic cellular grid (simulation time 30 h, growth according to the rules in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a> – <i>Model 9</i>), with both external and local (area-dependent) auxin sources (and sinks). Kinetic parameters:  = 100 (µm² min)⁻¹;  = 2.10⁷ min⁻¹. Figures S6, S7, S8, S9 illustrate the dependence of the shape of the auxin gradient on the parameters used here (in a non-growing root). More information on the kinetic equations can be found in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s016" target="_blank">Text S1</a>.</p

Cell-autonomous regulation: Robustness to perturbations.

<p>(A) Simulation output (time: 87 h) from a model as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g003" target="_blank">Figure 3</a>, yet with maximally ±25% noise added to cell cycle time (except first division ±10%, <i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a> – <i>Model 4</i>; see also Figures S3A and S3C), which demonstrates that the symplast can act as a stabilizing framework that dissipates ‘noisy’ mechanical perturbations through its structure. Colouring is according to growth potential (defined in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a> section) as a measure for ‘turgor pressure’. (B) Timer-based developmental transitions can amplify tissue distortion. Simulation output of timer mechanism with layer-specific cell cycle time (<i>Model 5</i> at simulation time 90 h). Upon release from the ‘QC’ cells can divide 3 times. For the outer 6 cell files CCT = 900 min, the inner 6 CCT = 1080 min. Afterwards, cells undergo accelerated growth until 4440 min after ‘QC’-release. Therefore the inner cells lag behind if their release started more or less synchronous as is the case here (maximally ±10% noise). Synchronicity is an important factor determining stable patterning in itself. Counters are sensitive as they multiply and therefore consistently amplify differences in CCT. Cells in the centre are lagging behind in terms of growth. As simulation time passes the irregularities eventually result in severely distorted patterns. Colouring is according to growth potential (defined in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a> section) as a measure for ‘turgor pressure’. (C) Simulation of <i>Model 6</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>) at simulation time 42 h, with a sizer-based cell cycle. One sizer, imposing division at a defined absolute cell size, is used despite differences in width of cells at similar positions along the main growth axis. Outer cell files have wider cells which reach the critical size before those of inner files. Therefore they undergo a much earlier exit from proliferation starting accelerated growth earlier, resulting in cell shape/tissue distortion. Cells in the centre are therefore lagging behind in terms of growth rate. Cell-autonomous regulatory systems appear inherently sensitive to this effect. Colouring is according to growth potential, GP or ‘turgor pressure’ (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>).</p

Layer-driven growth can alleviate problems with the ULSR.

<p>(A) Layer-driven auxin-dependent growth according to <i>Model 11</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910.s014" target="_blank">Table S1</a>). Simulation time 109.5 h of model for which auxin concentration is ‘interpreted’ by the two layers of border cells (in analogy with endodermis-specific growth regulation by GA <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi.1003910-UbedaToms2" target="_blank">[78]</a>, a different tissue layer, for instance the epidermis, could be equally effective: result not shown) and translated into an increase in the target area of those cells (<i>cf.</i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a>). The other cell layers are programmed to follow passively by re-setting their target areas to their actual areas after every simulation step in accordance with a small resisting force w.r.t. the layer that is controlling growth. Colouring is according to growth potential (‘GP’, as defined in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#s4" target="_blank">Methods</a> section) as a measure for ‘turgor pressure’, showing border cells drive growth of neighbouring cells to the extent that their target areas are smaller than their actual areas (slight blue colour). (B) Plot of root length versus simulation time shows steady linear organ growth from 94 h on after a long preparatory phase to construct a realistic starting grid with a stable auxin gradient (code details in Dataset 1). (C) Plot depicting the cell length along the principal growth axis at step 103.5 h of the simulation with a model equivalent to <i>Model 8</i> but with the growth driven by the 3th and 10th layer as in <i>Model 11</i>. Note that cell lengths vary smoothly from DZ to EZ similar to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g002" target="_blank">Figure 2</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003910#pcbi-1003910-g005" target="_blank">5C</a>.</p