Adequate IAA production from Escherichia coli could efficiently promote plant root growth. However, IAA is toxic to plants if its concentration is too high. Therefore it is important to predict the auxin expression level with a certain promoter, and then adjust the promoter strength to ensure the auxin produced by genetically modified (GM) E.coli will optimally increase root growth.

Auxin increases root growth in terms of root length and number of branches. In order to study how different concentrations of auxin effect root growth patterns, modelling tools were combined with wet lab results to predict and visualise Arabidopsis root growth in terms of root length and number of branches.

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2.1 Objective

Determine the auxin expression level of a single E. coli cell with auxin promoter strength 4.536 RNA/min/μg substrate DNA, and then predict the number of bacteria needed to be placed in the seed coat to induce optimal root growth.

2.2 Description

The genetically engineered auxin pathway involves two genes, IaaM and IaaH, both of which are constitutively expressed. The IaaM gene encodes tryptophan-2-monooxygenase (T-2-monase) that catalyses the conversion of tryptophan (Trp) to indole-3-acetamide (IAM), which is then hydrolysed to release indole-3-acetic acid (IAA) by the hydrolase iaaH [3]. At the same time the synthesised IAM and IAA will competitively inhibit the enzyme activity of tryptophan-2-monooxygenase, thereby inducing a negative feedback loop on the expression of IAA. The enzymatic reactions involved in the pathway are illustrated in Figure 1 below.

Fig.1 [1]: Auxin symtheis pathway

In addition, based on research carried out by the University of Colorado [4], tryptophan is also negatively controlled inside bacteria. Therefore the tryptophan synthesis pathway should be integrated into the above model. Furthermore, in order to reduce the numbers of parameters, as most of the parameters in Figure 1 are not available, the auxin pathway is then simplified into two Michaelis Menten equations, which are combined with the tryptophan pathway and constitutive gene expression for T-2-monase and iaaH. The whole tryptophan auxin pathway model was described in Equation 1 [5], and the parameters are defined in the parameters section below.

The assumptions associated with this model are listed below.

(1) We neglect the short delay due to synthesis of Trp-T-2-monase (substrate-enzyme (ES) complex), IAM-iaaH (ES complex), IAM-T-2-monase (inhibitor-enzyme (EI) complex) and IAA- T-2-monase (inhibitor-enzyme (EI) complex) and thus to consider these species reach their equilibrium almost instantaneously.

(2) The degradation rate of IAA is extremely low (7 days according to the experiments) compared to bacterial growth rate. Therefore we used bacterial growth rate as the IAA degradation rate in this model.

(3) During the modelling of the auxin pathway, we found that the rate determining species for IAA synthesis is IAM, not the enzyme iaaH, since the production of IAM is inhibited by itself and IAA.

2.3 Result and discussion

Figure 2 below represents the output of our model. It shows how the concentration of each protein species varies with time. It shows the simulation of the enzymatic reaction for each of the species, with initial concentration of O_F = 1.54×10^-4 µM, M_F = 3.78×10^-4 µM , E = 0.378µM , Trp = 4.1µM and all others = 0 [4].

Fig.2(a): The evolution of IAM vs. time. Fig.2(b): The evolution of IAA vs. time The graph Fig.2(b)) shows IAA concentration is 72.35 µM, therefore each bacterium produces 7.24×10^-14 µM at steady state with bacterial volume equals to 10^-15 dm³. From wet lab experiment we know that the optimal concentration of IAA to promote root growth is 0.1nM, and the volume of Arabidopsis seed with seed coating approximately equals to 4.2×10^-9 m³[7]. Therefore the number of bacteria required to maximally increase root growth of bacteria is 5, the value varies due to the variation of seed size of different of plant.

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3.1 Objective

1.Create a graphic program to demonstrate the growing process of the arabidopsis root system, based on the principles of Lindenmayer system and plant physiology.

2. Use Matlab data fitting tools to develope relations between auxin concentration and growth rate and number of branches.

3.2 Description

In order to using computational tools to visulise the root growth phenomena (primary rootlength, branching, root density, etc) in different enviromental conditions, root order and root length are considered. Root order describes the branching “generation” of a root system, a root without branching is defined as a zero-order root. And root growth depends on the environmental factors, such as gravitation, soil heterogeneities, etc.

3.2.1 Tropisms

A root system starts with a single root tip of a zero-order root. Then the root grows away from the plant stem in a conical way.[10]

Root growth depends on the environmental factors, such as gravitation, soil heterogeneities, etc. Therefore, two more variables are defined to describe the plant adaptation:[10]

α:-

how strong the roots direction changes per 1cm growth ?

larger value indicates a more deflected root and a more twisted root system

N:-

the number of trials for the roots to find the optimal angles α and β for the rotation

for the downward movement

N can be any real number, if N = 1.5, if means that N can be either 1 or 2.

Fig.3:The difference of the root systems with different values of N and σ

3.2.2 Lindenmayer system and root growth modelling

An L-system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms, due to the two main properties: recursive nature and self-similarity.[11] Plant models and natural-looking organic forms are easy to define, as by increasing the recursion level the form slowly 'grows' and becomes more complex.

Using L-systems for generating graphical images requires that the symbols in the model refer to elements of a drawing on the computer screen. It interprets each constant in an L-system model as a turtle command.

3.2.3 Root Growth

The modelling of auxin uptake will give prediction of the root system development in the following ways:-

"What is the primary root growth rate?"

"What does the root system look like after a certain period of time?"

"How does arabidopsis respond to different auxin concentration?"

... ...

We modified an MATLAB program developed by Daniel Leither[6] research group demonstrate the 3D root system based on the principles of Lindenmayer system(turtle commands) and the root growth modelling toolbox developed by Daniel Leitner et al from BOKU(Universität für Bodenkultur Wien, University of Natural Resources and Life Sciences, Vienna).[12]

3.2.4 Data fitting

Apart from the growth rate parameters we took from the literature, we analyzed the raw data from wetlab to give more accurate and suitable prameters for our own project.

When the arabidopsis samples are planted, we record the root length and number of branches every three day from day 0 to day 9. Then, root length, daily root growth rate and number of branches are plotted against time and auxin concentration.

3.3 Result and Discussion

3.3.1 Visualisation of a root system

The values from literature gives the relationship between external auxin concentration and elongation of the roots is 5*10^-5 mol/L → 200 µm elongation in 30 mins. The modelling parameter of growth speed is therefore 9.6*10^-3 m/day.[9]

By observing the real roots grow from the plant, the demonstration is modified to give a more reliable and accurate prediction of the root growth. Arabidopsis has a primary root with zeroth order and it is thicker than the branches. Arabidopsis normally grows to the depth of 20~30cm inside the soil and branches once only. The 3D picture shown below predicts the root growth with different elongation rate(with auxin = 0.96cm/day; without auxin = 0.46cm/day[10]). They can be compared with the photo of real root system.

Fig.4: 3D visalisation of the arabidopsis root system. The graph Fig.4(a)(b) shows gives the demonstration of the arabidopsis root system in different auxin concentration conditions with a growth time of 20days.Fig.4(c)(d) are the photos of real arabidopsis taken from literature and our wetlab.

Fig.5: The graphic analysis of the 3D root system. By taking cross and longitudinal sections of the root system at different positions, Fig.5(a) is the root length distribution(by fraction)against the root depth.Fig.5(c) is the surface area distribution map. This graph will give the guidance of placing GM E.coli inside the soil. E.coli should be placed at the depth with the maximal root surface area(6cm) inside the soil for the maximal uptake efficiency.

The root has a growth rate of 0.96cm/day with the external auxin concentration 5x10-5mol/L, however, this data is selected from literature. To get an accurate growth rate which is particularly fitting our project, we decided to do data fitting analysis to the arabidopsis we plant.

3.3.2 Data fitting

The data fitting plots are analysed to give an approximation of the relationship between auxin concentration and root growth. The following graph gives an example of root length against time. [8]

Fig.6: Root growth speed decays against time. This graph gives a prediction of the root growth speed (cm/day) for 20 days. The exponential decay constant is 0.048.

Fig.7: Primary root length(mm) vs. time(day) and external auxin concentration(mol/L). The data fitting result gives this plot of the arabidopsis primary root length. The length increases as the growth time increases, and reaches to a maximal length into the soil when the auxin concentration is approximately 1pM.

Fig.8: Primary root growth rate(mm/day) VS root growth time(day) We used the data fitting toolbox of Matlab to obtain this figure. The relationship between the growth rate and the auxin concentration can be approximated by Gaussian equation. The abnormality of the 0.1nM curve is due to the two contaminated samples which stopped growing at 7mm after Day 5. Fig.8 is consistent with the prediction of the decay of the root growth speed given by Fig.6.

Fig.9: Number of lateral branch VS external auxin (log)concentration The optimal concentration for lateral root branching = 1uM-10nM, at this concentration, the arabidopsis root gained the most lateral branches .

The result of data fitting does not coincide with our latest wetlab experiment results, since the optimal external concentration for arabidopsis is found to be 0.1 nM. The failure of the previous experiment is analysed and gives the guidance for the new experiments in the following ways:

1. Instead of measuring the primary root length and number of lateral branches of arabidopsis, the growth rate should be tested by measuring the dry mass of the roots.

2. At the early stage of planting, the number of lateral branches is very small, therefore massive percentage errors and standard derivation are produced.

3. Huge error is also introduced by the dilution process of the auxin solution to reach a low concentration level of 10^{-13 mol/L.}

4. The wetlab result shows that the auxin concentration in phytogel media is negligible after 9 days, the degradation of auxin in an important factor when the root growth is modelled.

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