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Putting theory to the test: which regulatory mechanisms can drive realistic growth of a root?

De Vos D, Vissenberg K, Broeckhove J, Beemster GT - PLoS Comput. Biol. (2014)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Molecular Plant Physiology and Biotechnology, Department of Biology, University of Antwerp, Antwerp, Belgium.

ABSTRACT
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 Arabidopsis thaliana 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.

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Auxin dynamics on a dynamic cellular grid.Output of model simulations (Table S1 – Model 9) with cells growing exponentially (in accordance with [32]) 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 Text S1, Figure S5). (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 (cf.Text S1):  = 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 Table S1 – Model 9), 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 Text S1.
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pcbi-1003910-g006: Auxin dynamics on a dynamic cellular grid.Output of model simulations (Table S1 – Model 9) with cells growing exponentially (in accordance with [32]) 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 Text S1, Figure S5). (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 (cf.Text S1):  = 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 Table S1 – Model 9), 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 Text S1.

Mentions: A consequence of the symplastic growth of the root is that at a given distance from the tip all cells have the same relative expansion rate [32]. As stated by Ivanov [33], any observed difference in cell lengths between tissues must therefore reflect differences in cell proliferation (see also [26]). Inversely, any form of growth regulation that results in different elongation rates for cells at the same distance from the tip would disrupt symplastic growth (Figure 1A). For instance, suppose all cells at the same (vertical) position in a downward growing root have the same absolute (areal) expansion rate, irrespective of their size (Model 1, Tables 1 and S1). With inner cell files narrower than outer cell files (similar to the real root) this fixed size increment results in consistently larger relative elongation rates for the inner tissue layers leading to tissue distortion and unbalanced distribution of mechanical stresses (Figure 1B and C). Note that the same situation would occur when adjacent files contain cells of similar width, but different lengths growing at the same absolute rates. Hence, non-uniform relative strain rates at some position along the principal growth axis eventually lead to malformations.


Putting theory to the test: which regulatory mechanisms can drive realistic growth of a root?

De Vos D, Vissenberg K, Broeckhove J, Beemster GT - PLoS Comput. Biol. (2014)

Auxin dynamics on a dynamic cellular grid.Output of model simulations (Table S1 – Model 9) with cells growing exponentially (in accordance with [32]) 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 Text S1, Figure S5). (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 (cf.Text S1):  = 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 Table S1 – Model 9), 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 Text S1.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4214622&req=5

pcbi-1003910-g006: Auxin dynamics on a dynamic cellular grid.Output of model simulations (Table S1 – Model 9) with cells growing exponentially (in accordance with [32]) 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 Text S1, Figure S5). (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 (cf.Text S1):  = 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 Table S1 – Model 9), 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 Text S1.
Mentions: A consequence of the symplastic growth of the root is that at a given distance from the tip all cells have the same relative expansion rate [32]. As stated by Ivanov [33], any observed difference in cell lengths between tissues must therefore reflect differences in cell proliferation (see also [26]). Inversely, any form of growth regulation that results in different elongation rates for cells at the same distance from the tip would disrupt symplastic growth (Figure 1A). For instance, suppose all cells at the same (vertical) position in a downward growing root have the same absolute (areal) expansion rate, irrespective of their size (Model 1, Tables 1 and S1). With inner cell files narrower than outer cell files (similar to the real root) this fixed size increment results in consistently larger relative elongation rates for the inner tissue layers leading to tissue distortion and unbalanced distribution of mechanical stresses (Figure 1B and C). Note that the same situation would occur when adjacent files contain cells of similar width, but different lengths growing at the same absolute rates. Hence, non-uniform relative strain rates at some position along the principal growth axis eventually lead to malformations.

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Molecular Plant Physiology and Biotechnology, Department of Biology, University of Antwerp, Antwerp, Belgium.

ABSTRACT
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 Arabidopsis thaliana 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.

Show MeSH
Related in: MedlinePlus