Institute of Cytology and Genetics, SB RAS, Lavrentyeva 10, Novosibirsk, Russia

Novosibirsk State University, Pirogova 2, Novosibirsk, Russia

Institute of Mathematics, SB RAS, Koptjuga 4, Novosibirsk, Russia

Department of Computer Science, University of California, Irvine, USA

Institute for Genomics and Bioinformatics, University of California, Irvine, USA

Abstract

Background

In plant roots, auxin is critical for patterning and morphogenesis. It regulates cell elongation and division, the development and maintenance of root apical meristems, and other processes. In

Results

We propose a

Conclusion

We present here a plausible mechanism for auxin patterning along the developing root, that may provide for self-organization of the distal auxin maximum when the

Background

Plant architecture is formed by the activities of meristems, which comprise stem cells and their derivatives, giving rise to various cell types. The root apical meristem (RAM) is formed at the earliest stages of embryogenesis and is localized to the root apex after germination

Auxin concentration maxima in plant tissues are mainly formed due to active auxin transport between cells

Understanding of the role of active transport in the formation of auxin concentration gradients is a topical problem in developmental biology. Computer modeling approaches are also used for solving this problem. In particular, Jonsson et al. (2006), Smith et al. (2006), and de Reuelle et al. (2006) have studied models for phyllotaxis mechanisms in the shoot

The

In a number of works, it has been shown that the formation of an auxin gradient precedes tissue patterning (reviewed in

Here, we present a mathematical model that implements the

We will demonstrate that the

Methods

Simulation of Auxin Distribution in Root

Biological assumptions

The acropetal transport is of dominant importance in supplying auxin to the root in the early stages of seedling development, until the root gains competence to synthesize its own auxin about 5 days after germination

Of all the auxin carriers, we currently only consider PIN1. PIN1 is expressed in the root vasculature and weak expression is sometimes observed in the QC

It has been shown that at low concentrations, auxin activates transcription of

Mathematical description

I. Elementary processes in the model of auxin distribution

Auxin redistribution along the central root axis is described based on the interactions of a limited set of dynamic intracellular processes, namely:

The rate of auxin flux into the cell is described by the following rate equation:

where _{a }

The conjugation, oxidation, and lateral distribution of auxin are summarized in the generalized degradation process. The rate of auxin degradation in the cell is described as

where _{d }

The rate of auxin diffusion from one cell to another is described as

where

The rate of active transport via the PIN1 proteins (

where _{0 }

PIN1 concentration dynamics in individual cell was defined by auxin-dependent rates of PIN1 synthesis and degradation (Figure

Representation in the model the processes influencing the auxin distribution along the central root axis

**Representation in the model the processes influencing the auxin distribution along the central root axis**. a. Acropetal flow is considered in the model along the cell array on the central root axis (

This function is zero at zero auxin concentration in a cell and monotonically increases to

The rate of PIN1 protein degradation is modelled as the rational polynomial, monotonically increasing as a function of its arguments

In Eqs. (5) and (6) the parameter _{1 }
_{1 }
_{2 }
_{2 }
_{3 }
_{
1
}and _{2 }

The biological and mathematical description of PIN1 polarization mechanisms is an important scientific challenge, but it is beyond the scope of this work. We consider an array of functionally identical cells, where PIN1 proteins are polarly localized at one cell side, mediating active auxin transport only in the acropetal direction.

II. Description of the 1

The one-dimensional (1

Auxin from the shoot first enters the

Diffusion is regarded as an isotropic process of auxin movement from the current cell to both preceding and next cells. Eq. (3) is used to describe the diffusion along the central root axis. For cell 1, passive diffusion is necessarily defined only to cell 2 due to the physical boundary conditions - there being no adjacent cell in the other direction. For cell

We consider that it is possible for PIN1 protein to be synthesized and degraded in each cell of the array, in an auxin concentration-dependent manner. (Figure

Summarizing all these assumptions, we get the following system (hereinafter, the 1

In model (7), _{i }
_{i }

The two-dimensional (2_{
j, i
}and _{
j, i
}, respectively. Every inner layer of the cell layout for varying index _{
i
}and _{
i
}are changed to _{
j, i
}and _{
j, i
}for the

The cell layouts in the 2D models of auxin distribution

**The cell layouts in the 2D models of auxin distribution**. a. The cell layout in the

The outer "epidermal" layers,

where _{1 }= +1, _{M }

**The Models details**. The supplementary text containing the following chapters:

Click here for file

Comparing the cell layout described in Grieneisen et al., (2007)

Modeling of Root Cell Growth and Division

Biological Background

The profile of cell mitotic activity along the meristematic zone of the root is bell-shaped with the maximum located at a distance of 10-16 cells from the QC

Mitotic activity in the root and its simulation

**Mitotic activity in the root and its simulation**. a. The scheme of root tip structure in Arabidopsis. The cells of different types marked by different colors (for details, see figure 1). b. Qualitative profile of mitotic activity in cells along the central root axis. Two maxima of mitotic activity are distinguished along the central root axis according to Dolan et al. (1993)

Cell divisions in root are governed by different hormones. Depending on the concentration, auxin acts in different fashions on the cell division rate: low and high auxin concentrations have a negative effect, whereas its intermediate concentrations has a positive effect

Mathematical description

I. Dynamical grammar for more realistic cell dynamics simulation

To find out whether a more realistically cellularized model that includes cell growth and cell division dynamics might disrupt or destroy the pattern formation process, the 1

II. Description of cell cycle in the model

The cell cycle in the model is described in the formalism of DG stochastic rules as a sequence of two phases: growth phase and idle phase. The moments of growth and idle phase completion are random variables with probability density functions depending on the functions _{GP }
_{IP}

Increase in cell length

During the idle phase the cell does not grow, so _{
0
}+_{
growth
}

The growth phase in the cell commences immediately at time _{GP}(r)

where _{min }
_{GP }(r)

The idle phase duration defines the rate of cell division. In the model the function of the idle phase completion _{IP}

The values of parameters in Eq. (12) are selected so that the idle phase is be longer in case of either deficiency or excess in the _{0 }

II. Formation of the gradient of

To simulate a realistic cell dynamics along the central root axis (Figure

Synthesis of the _{
DivF
}(_{
i
}, _{
i+1}) depends on the difference in auxin concentration in the cell compared to the (

where constant _{s, DivF}
_{
i
}, _{
i+1}) grows monotonically in the interval [0;1] as the _{
i+1 }-_{
i
}gradient increases. Due to the boundary position of the cell

The rate of _{d, DivF}

where _{3 }
_{4 }

The rate of

Where _{DivF }

The dynamics of

In Eq. (16), _{i }

Practical aspects of modeling

Numerical solution

The nonlinear system of equations of 1

**The 1 D extended model in Mathematica**. The Mathematica file implementing the 1

Click here for file

Parameter estimations

The parameters were estimated using published experimental data on the mechanisms involved in the regulation of

Verification of model calculation results

The proposed model is based on published data. The experimental data on the auxin distribution in the root are represented by images of roots from transgenic

**Processing of the experimental data on auxin distribution in root**. The figure showing the applied method of the experimental data on auxin distribution from DR5 auxin response images conversion to relative auxin concentrations in root cells. In the figure, (a) auxin distribution in the root according to DR5 reporter activity from Sabatini et al. (1999)

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Results

Mechanisms of Auxin Acropetal Transport Regulation are Sufficient for Self-organization of Auxin Distribution Pattern in Roots

A characteristic feature of auxin distribution patterns in the root is the presence of concentration maximum in the root cap initial cells (Figure

The auxin distribution pattern reproduced by the model

**The auxin distribution pattern reproduced by the model**. a. Expression of DR5::GUS detects the auxin pattern in the root tip (adapted from Sabatini et al., 1999;

**The model parameters**. The table containing all model parameters.

Click here for file

The mechanism of auxin distribution self-organization found in the resulting stationary solutions is the following. In cells with low auxin concentration, the positive regulation of PIN1 expression by auxin provides for self-enhancement of the acropetal auxin flow. This results in a rapid auxin accumulation at the root end (cell number 1). This is followed by the increase of auxin concentration in the neighboring cell 2 whereto high amount of auxin moves by diffusion from the cell 1 (reflected flow of diffusion). As soon as the auxin concentration in cell 2 exceeds the threshold for auxin-dependent PIN1 degradation (_{3}

The

Maintenance of auxin maximum in a growing root

To investigate whether cell divisions might disrupt or destroy the auxin distribution pattern that arises under the _{
α
}) The root growth in the 1 D _{
α
}. This effect can be avoided by adding to the model processes of self-regulated auxin synthesis (data not shown). Consistent with this

**Simulation of root growth along the central root axis**. In the figure, (a) Distribution of auxin (red), Y (blue) and rates of cell division (gray columns) in conventional units along the central root axis. The curves were calculated in the 1 D

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**Maintenance of auxin maximum at the root tip in the in silico growing root under normal condition**. The movie 1 simulated using the 1D

Click here for file

Root Apical Meristem Patterning Along the Central Root Axis

An additional morphogen was introduced to the 1

Consequently, we obtain

Thus, the positional information established

Simulation of Auxin Distribution under Various Conditions

The model analysis has demonstrated that the distal auxin maximum, found in the fifth cell of provascular layers, is formed for different sets of parameters and the model tolerates their slight variation. However, more significant changes in the parameter values led to a shift in the maximum position. Moreover, additional maxima of auxin concentration appear along the root axis [Additional files

**Analysis of the 1 D minimal model with basic set of parameters**. The figure showing changes in the auxin distribution pattern at N = 50 in response to variations in: (a-b).

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**Analysis of the 1 D minimal model with robust set of parameters**. The figure showing changes in the auxin distribution pattern at N = 50 in response to variations in: a.

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Increasing auxin flow from the shoot results in formation of additional inner maxima

Under favorable conditions, the acropetal auxin flow from the aboveground part of developing plant constantly increases _{
α
}) changes exogenously in time as _{
α
}= _{0 }+ _{
α
}is insufficient to compensate for the dilution of auxin concentration in the root caused by its growth (which is reached through increase in cell size and number). Consequently, the total auxin concentration in the root decreases, thereby leading first to the shift of the distal auxin maximum to the root end, _{1 }≈ _{2 }≈... ≈_{
N
}. On the other hand, if a high growth in the auxin flow from the shoot to the root is specified, then the total auxin content in the

**Periodic formation of additional inner auxin maximum in the in silico growing root under increased rates of auxin flow from the shoot**. The movie 2 simulated using the 1

Click here for file

We used the 1_{
α
}was decreased to 0.01 _{
α
}. rate. At _{
α
}= 0.45, an auxin concentration maximum in the stationary solution was formed in the cell 2 and the subsequent increase in _{
α
}determined the shift of this maximum towards the root base. In particular, the distribution formed at _{
α
}= 1 fits the experimental distribution (Figure _{
α
}= 1.2, the maximum appears in cell 11 (Figure _{
α
}rate had a critical value, _{
α
}~ 1.23, at which the stationary solution lost its stability. As a result, continuous oscillations of intracellular auxin concentrations appear in the model with a high-amplitude zone in the middle of the root (Figure _{
α
}value (

Sensitivity of the auxin distribution pattern to parameter and initial data variations

**Sensitivity of the auxin distribution pattern to parameter and initial data variations**. a.-c. The 1_{0 }(c.) that model changes in auxin flow from the shoot and treatment with auxin transport inhibitors, respectively. Oscillatory solutions of the model marked by dashed lines. b. When auxin flow from the shoot is too high, we observe unstable fluctuations of auxin concentration in the middle of the root. The curves correspond to unstable solutions calculated at equal time intervals. d.-f. The 1D

The effect of active auxin transport inhibitors on the auxin distribution pattern in the root

Treatment of the root with auxin active transport inhibitors, such as NPA or TIBA, leads to smearing of the auxin distribution pattern in the RAM _{0 }
_{0 }
_{0 }
_{0 }
_{3}

Self-restoration of auxin maximum and RAM structure after root tip cutting or QC ablation

After cutting off the root tip or ablating the QC by laser, the RAM gradually returns to its normal structure by forming a new QC from vascular initials _{j, i }

The auxin transport system can buffer changes in auxin concentration in localized tissues

IAA treatment of _{i }

Different sets of parameter values can be realized in different plant species

The mechanisms of auxin transport in higher plants are highly conserved _{2 }= 10 specifies a pronounced inhibition of the PIN1 expression when intracellular auxin concentration reaches the threshold value _{3 }

**Comparison of the models behavior with basic and robust sets of parameters**. The table showing the differences in the

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Discussion

In this work, we propose and substantiate a plausible mechanism for self-organization of the auxin distribution pattern along the central root axis, and its effect on establishment of the RAM. The

The mechanisms of auxin distribution pattern formation

At least four main mechanisms of auxin distribution pattern formation have been suggested. The

Another well-known mechanism that can describe auxin distribution patterning but hasn't been implemented for this purpose before is the

The Reverse Fountain Mechanism vs The Reflected Flow Mechanism

In this study we have shown that the

There are at least three effects provided exclusively by the

The main advantage of the

By comparison with the

We suggest that the

A Plausible Mechanism of Rhizotaxis

During plant growth, the acropetal auxin flow from the aboveground plant part to the root increases during development and plays a key role in regulating elongation of the main root and development of lateral roots

How Acropetal Auxin Flow Determines Cell Fate Specification Along The Central Root Axis

In the 1

The same profile of mitotic activity along the central root axis could be observed (data not shown) if we replace

Simulation of Auxin Transport in Different Plant Species

Auxin transport in plants is a highly conserved mechanism: auxin distribution patterns with the maximum in the stem cells of the RAM have been demonstrated for

First, the model with the robust set of parameters is sensitive to auxin fluctuations--additional auxin concentration maxima appeared in the middle of the root and at its base in simulations of root treatment with rather low doses of exogenous auxin. Moreover, once formed, these additional maxima were stably retained during root growth in the 1

Analysis of the model's behavior using two sets of parameters suggests a key role for the auxin transport inhibition mechanisms in the formation of different root system types. The model's behavior with the basic set (low efficiency of auxin-dependent inhibition of

Conclusions

In this work, we have studied

Authors' contributions

VVM, EM, VAL and NAO wrote the manuscript. VVM carried out the majority of the model design, calculation, analyses and the results interpretation. NAO and NAK contributed to the design and planning of the simulations, participated in data analysis. GY created the model with cell dynamics and performed its analysis. SIF performed the model calculation and analysis. EM developed the model with cell dynamics, and contributed to the design and planning of the simulations. VAL developed the mathematical model, conceived of the study, participated in its design and data interpretation. All authors read and approved the final manuscript.

Acknowledgements

We thank Tigran Bacarian, Petr Ponomarenko, Katerina Novoselova and Victoria Lavreha for critical reading of the text; Elliot M. Meyerowitz, Marcus Heisler, Adrienne Roeder (Caltech, USA) for their valuable advice and comments on this work. This research was supported in part by the US NSF (PHY05-51164 and EF-0330786) US NIH (P50 GM76516), RFBR (

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