"

From 2011.igem.org

Modelling

Introduction

Diverse bacterial species possess the ability to produce the auxin(indole-3-acetic acid, IAA, phyto-hormone). Different biosynthesis pathways have been identified and redundancy for IAA biosynthesis is widespread among plant-associated bacteria.[1] A high degree of similarity between IAA biosynthesis pathways in plants and bacteria was observed. [1]

The identification of intermediates led to the identification of five different pathways using tryptophan as a precursor for IAA. The synthesis pathway we used in our project consists of two steps with an intermediate of IAM(indole-3-acetamide). [2]

tryptophan-IAM(IaaM gene - tryptophan-2-monooxygenase)-IAM-IAA (IaaH gene-IAM hydrolase)

With the study on the auxin production process, we can predict the response of the plant (Arabidopsis) root system to the introduced auxin by generating a visualisation program of root systems.

Objectives

1. In order to choose the appropriate RNA promoter with the optimal strength, the auxin production amount is modelled base on the pathway via the intermediate IAM from the precusor tryptophan. The result of modelling answers the question: " How much auxin can be produced by the genetically modified bacteria with a typical RNA promoter strength? "

2. A graphic program is created to demonstrate the growing process of the Arabidopsis root system, based on the principles of Lindenmayer system and plant physiology.

Description

Auxin Synthesis Pathway

tryptophan-IAM(IaaM gene - tryptophan-2-monooxygenase)-IAM-IAA (IaaH gene-IAM hydrolase)

This pathway has a feedback inhibition mechanism, the production of IAM and IAA inhibits the function of tryptophan-2-monooxygenase, therefore stops the reaction chain. The inhibition of the enzyme activity is competitive, as shown by the equillibium equations below.

E + S ↔ ES → E + P

E + I ↔ EI

Also, the reaction kinetics fits the Michaelis-Menten kinetics model perfectly.

To write up the modelling code,a set of ODEs describing the concentration changes of each chemicals can be used to define the reaction process:

The rate constants of the reactions inside the pathway are required (parameters required: k1,k-1,k3,k-3). All the parameters of the two enzymes involved in this pathway, tryptophan-2-monooxygenase and IAM hydrolase, can be found at the enzyme database Brenda. [2]

Auxin uptake and root growth

1. Initial a root system

To visualise our modelling result, a root system is demonstrated to show the root growth phenomena(primary root length, branching, root density, etc) in different environmental conditions(external and internal auxin concentration).

1.1. Root order:-

Root order describes the branching “generation” of a root system, a root without branching is defined as a zero-order root

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.[3]

Fig.1 a conical approximation of the root system

1.2. Tropisms:-

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

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.2 the difference of the root systems with different values of N and σ

2. Lindenmayer system and root growth modeling

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.[3]

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.

L-systems are now commonly known as parametric L systems, defined as a tuple.

A production consists of two strings, the predecessor and the successor.

For any symbol A in V which does not appear on the left hand side of a production in P,the identity production A → A is assumed. These symbols are called constants or terminals.

An L-system is context-free if each production rule refers only to an individual symbol and not to its neighbors. Context-free L-systems are thus specified by either a prefix grammar, or a regular grammar.

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. 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 wrote a program using Matlab to draw 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).

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.

Results

Auxin Synthesis Pathway

blablablabla...Si this is your part

Auxin uptake and root growth

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.

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 = 0.46cm/day; without auxin = 0.96cm/day). They can be compared with the photo of real root system.

Fig.4 visalisation of the arabidopsis root system

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.

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.

Fig.5 root growth speed decays against time

Fig.6 primary root length(mm) VS time(day) and external auxin concentration(mol/L)

Fig.7 primary root growth rate(mm/day) VS root growth time(day)

Fig.8 number of lateral branch VS external auxin (log)concentration

From Fig.6 and Fig.8, the following conclusion can be make:-

the optimal concentration for primary root growth = 1pM, at this concentration,

the arabidopsis root reached the maximal depth into soil

the optimal concentration for lateral root branching = 1uM-10nM,

at this concentration, the arabidopsis root gained the most lateral branches

We used the data fitting toolbox of Matlab to obtain Fig.7 primary root growth rate(mm/day) VS root growth time(day), 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.7 is consistent with the prediction of the decay of the root growth speed given by Fig.7.

Matlab code

Reference