Team:Imperial College London/Project/Auxin/Results/Modeling

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<h1>Modeling</h1>
 +
<h2>Auxin  synthesis pathway</h2>
 +
<p>    The production of auxin by bacteria(E.Coli and/or B.Subtilis) is one main module in our project. 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.
 +
<p>    The result of modelling answers the question: <i>" How much auxin can be produced by the genetically modified bacteria with a typical RNA promoter strength? "</i>
 +
<p>    the pathway has two steps:[1]
 +
<p>  <em>tryptophan-IAM(IaaM gene -  tryptophan-2-monooxygenase)</em><em>-</em><em>IAM-IAA (IaaH  gene-IAM hydrolase)</em><br />
 +
<p>    A feedback inhibition mechanism exists in the pathway, the  production of IAM and IAA inhibits the function of tryptophan-2-monooxygenase,  therefore stops the reaction chain</p>
 +
<br>
 +
<ul>
 +
      <b><li>competitive inhibition</li></b>
 +
</ul>
 +
<p>  E + S &harr; ES &rarr; E + P <br />
 +
<p>  E + I &harr; EI<strong></strong></p>
 +
<br>
 +
<ul>
 +
      <b><li>the reaction kinetics fits the Michaelis-Menten kinetics model perfectly <strong></strong></li></b>
 +
</ul>
 +
<p>    a set of ODEs can  be used to model the reaction process:
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/1/13/Equ.png" width="360" height="270" />
 +
<br>
 +
<ul>
 +
      <b><li>parameters required: k1,k-1,k3,k-3 </li></b>
 +
</ul>
 +
<p>    The rate constants of the reactions inside the pathway are required. All the parameters of the two enzymes involved in this pathway, tryptophan-2-monooxygenase and IAM hydrolase, can be found at the enzyme database <i>Brenda</i>. [2]
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/7/71/ICL_Auxin1.png" alt="" width="359" height="271" /></p>
 +
 +
<h2>Initial a root system</h2>
 +
<p>    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).
 +
<br>
 +
<ul>
 +
      <b><li>Root order:- </li></b>
 +
</ul>
 +
<p>    Root order describes the branching &ldquo;generation&rdquo; of a root system, a root  without branching is defined as a zero-order root</p>
 +
<p>    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]
 +
<p>    <img src="https://static.igem.org/mediawiki/2011/0/02/ICL_Auxin2.png" alt="" width="175" height="131" />
 +
      <img src="https://static.igem.org/mediawiki/2011/7/76/ICL_Auxin3.png" alt="" width="200" height="122" />
 +
<p>    initial number of  root = n<sub>0</sub><br />
 +
<p>    initial cone base  radius = r<sub>0</sub><br />
 +
<p>    axial insertion  angle = &alpha; <br />
 +
<p>    radial insertion angle = &beta;</p>
 +
<p>    To ensure a even distribution of the roots, &alpha; and &beta; must be set in the following way:
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/0/06/ICL_AuxinEqn6.png" alt="" width="119" height="22" /> <br />
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/3/35/ICL_AuxinEqn7.png" alt="" width="57" height="19" /> <br />
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/6/60/ICL_AuxinEqn8.png" alt="" width="71" height="19" /></p>
 +
 +
<br>
 +
<ul>
 +
      <b><li>Tropisms </li></b>
 +
</ul>
 +
<p>    Root growth depends on the environmental factors, such as gravitation,  soil heterogeneities, etc.
 +
<p>    Therefore, two more variables are defined to describe the plant  adaptation:
 +
<p>    <b>&alpha;:-</b>
 +
<p>    how strong the roots direction changes per  1cm growth ?
 +
<p>    larger value indicates a more deflected root and a more twisted root system
 +
<p>    <b>N:-</b>
 +
<p>    the number of trials for the roots to find the optimal angles &alpha; and &beta; for the rotation
 +
<p>    for the downward movement
 +
<p>    N can be any real number, if N = 1.5, if means that N can be either 1 or 2.</p>
 +
<br>
 +
<ul>
 +
      <b><li>the difference of the root systems with different values of N and σ can be shown </b>
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/4/4a/ICL_Auxin4.png" alt="" width="491" height="249" />
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/8/89/ICL_Auxin5.png" alt="" width="521" height="259" /></li>
 +
</ul>
 +
 +
<h2>Lindenmayer system and root growth modeling</h2>
 +
<p>    The properties of L-system provides the basic graphic principles to "draw" a root system.
 +
<p>    The  method to model root growth is to create a root system using Matlab.
 +
<br>
 +
<p>    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.[4]
 +
<p>      <i>-recursive nature</i>
 +
<p>      <i>-self-similarity </i>
 +
<br>
 +
<p>    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.
 +
<br>
 +
<p>    L-systems are now commonly known as <em>parametric</em> L systems, defined as a tuple.
 +
<p>    G = (<em>V</em>, ω, <em>P</em>)<br />
 +
<p>    V = a set of symbols containing elements that can be  replaced (<em>variables</em>)<br />
 +
<p>    ω (<em>start</em>, <em>axiom</em> or <em>initiator</em>) = a  string of symbols defining the initial state of the system
 +
<p>    P = a set of production rules defining the way variables can be replaced with combinations of constants and other variables.
 +
<p>    A production consists of two strings, the <em>predecessor</em> and the <em>successor</em>. <br />
 +
<p>    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 <em>constants</em> or <em>terminals</em>. </p>
 +
<br>
 +
<p>    An L-system is <em>context-free</em> 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.
 +
<p>    If there is  exactly one production for each symbol, then the L-system is said to be <em>deterministic</em> (a deterministic context-free L-system is popularly called a <em>D0L-system</em>).
 +
<p>  If there are several, and each is chosen with a certain probability during each iteration, then it is a <em>stochastic</em> L-system.</p>
 +
<br>
 +
<ul>
 +
      <b><li>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.</li></b>
 +
</ul>
 +
 +
<h2>Auxin uptake</h2>
 +
<p>    The modelling of auxin uptake will give prediction of the root system development in the following ways:-
 +
<p>    <i>"What is the primary root growth rate?"</i>
 +
<p>    <i>"What does the root system look like after a certain period of time?"</i>
 +
<p>    <i>"How does arabidopsis respond to different auxin concentration?"</i>
 +
<p>    ... ...
 +
<br>
 +
<p>    The auxin distributed in the soil enters the plant mainly by diffusion, if the convection process is neglected, then  the diffusion can by described using the following equations (Barber 1995)<br />
 +
  <img src="https://static.igem.org/mediawiki/2011/c/c7/ICL_AuxinEqn9.gif" alt="" width="521" height="39" /><br />
 +
<p>  &theta; = volumetric water content of the soil = 0.4<br />
 +
<p>  b =  buffer power = 100<br />
 +
<p>  c = auxin concentration<br />
 +
<p>  D<sub>1</sub> = diffusion coefficient of auxin<br />
 +
<p>  f =  impedance factor = 0.3<br />
 +
<p>  s =  root surface area per unit volume <br />
 +
<p>  F<sub>m</sub> = maximal influx = 2.5*10<sup>-7 </sup>&micro;mol/cm<sup>2</sup>/s <br />
 +
<p>  K<sub>m</sub> = Michaelis-Menten constant = 4*10<sup>-4 </sup>&micro;mol*cm<sup>3</sup><br />
 +
<p>  t<sub>age</sub> = average root surface  age <br />
 +
<p>  k = decay factor of auxin uptake</p>
 +
<p>    An anxin  distribution map can be drawn from the equation above.</li>
 +
<br>
 +
<p>    The  values from literature gives the relationship between external auxin concentration and elongation of the roots:-
 +
<p>    5*10<sup>-5 </sup>mol/L &rarr; 200 &micro;m elongation in 30 mins <br />
 +
<p>    The modelling parameter of growth speed is therefore 9.6*10<sup>-3</sup> m/day</p>
 +
<p>    use  L-system and turtle command, a zeroth-order root system is demonstrated</li>
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/6/67/ICL_Auxin6.jpg" alt="" width="295" height="329" /></p>
 +
 +
<br>
 +
<p>    As you can see, this demonstration has a more generalised root shape. Arabidopsis, which does not grow in this way,is used in our project. 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.
 +
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/thumb/0/0c/Ttt.png/800px-Ttt.png" alt="" width="800" height="644" /><br />
 +
<br>
 +
<p>  <object style="height: 390px; width: 640px"><param name="movie" value="http://www.youtube.com/v/rUsxWr9_cvg?version=3"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://www.youtube.com/v/rUsxWr9_cvg?version=3" type="application/x-shockwave-flash" allowfullscreen="true" allowScriptAccess="always" width="640" height="390"></object>
 +
<br>
 +
<p>    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.
 +
<ul>
 +
<p>  <b><li>data fitting:-</li></b>
 +
</ul>
 +
<p>    When the arabidopsis samples are planted, we record the root length and number of branches every day from day 0 to day 20.Then, root length, root growth rate and number of branches are plotted against time and auxin concentration. These three curves are analysed to give the best mathematical equation to describe it, This can be an approximation of the relationship between auxin concentration and root growth. The following graph gives an example of root length against time.
 +
<p>  <img src="https://static.igem.org/mediawiki/2011/7/7e/Plot1.PNG" alt="" width="506" height="582" /><br />
 +
 +
</body>
 +
</html>

Revision as of 11:27, 31 August 2011



Modeling

Auxin synthesis pathway

The production of auxin by bacteria(E.Coli and/or B.Subtilis) is one main module in our project. 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? "

the pathway has two steps:[1]

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

A feedback inhibition mechanism exists in the pathway, the production of IAM and IAA inhibits the function of tryptophan-2-monooxygenase, therefore stops the reaction chain


  • competitive inhibition
    •

E + S ↔ ES → E + P

E + I ↔ EI


  • the reaction kinetics fits the Michaelis-Menten kinetics model perfectly

a set of ODEs can be used to model the reaction process:


  • parameters required: k1,k-1,k3,k-3

The rate constants of the reactions inside the pathway are required. 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]

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).

  • 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]

initial number of root = n0

initial cone base radius = r0

axial insertion angle = α

radial insertion angle = β

To ensure a even distribution of the roots, α and β must be set in the following way:




  • 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.


  • the difference of the root systems with different values of N and σ can be shown

Lindenmayer system and root growth modeling

The properties of L-system provides the basic graphic principles to "draw" a root system.

The method to model root growth is to create a root system using Matlab.

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.[4]

-recursive nature

-self-similarity

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.

G = (V, ω, P)

V = a set of symbols containing elements that can be replaced (variables)

ω (start, axiom or initiator) = a string of symbols defining the initial state of the system

P = a set of production rules defining the way variables can be replaced with combinations of constants and other variables.

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.

If there is exactly one production for each symbol, then the L-system is said to be deterministic (a deterministic context-free L-system is popularly called a D0L-system).

If there are several, and each is chosen with a certain probability during each iteration, then it is a stochastic L-system.


  • 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.
    •

Auxin uptake

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?"

... ...

The auxin distributed in the soil enters the plant mainly by diffusion, if the convection process is neglected, then the diffusion can by described using the following equations (Barber 1995)

θ = volumetric water content of the soil = 0.4

b = buffer power = 100

c = auxin concentration

D1 = diffusion coefficient of auxin

f = impedance factor = 0.3

s = root surface area per unit volume

Fm = maximal influx = 2.5*10-7 µmol/cm2/s

Km = Michaelis-Menten constant = 4*10-4 µmol*cm3

tage = average root surface age

k = decay factor of auxin uptake

An anxin distribution map can be drawn from the equation above.

The values from literature gives the relationship between external auxin concentration and elongation of the roots:-

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

use L-system and turtle command, a zeroth-order root system is demonstrated


As you can see, this demonstration has a more generalised root shape. Arabidopsis, which does not grow in this way,is used in our project. 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.




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.

  • data fitting:-

When the arabidopsis samples are planted, we record the root length and number of branches every day from day 0 to day 20.Then, root length, root growth rate and number of branches are plotted against time and auxin concentration. These three curves are analysed to give the best mathematical equation to describe it, This can be an approximation of the relationship between auxin concentration and root growth. The following graph gives an example of root length against time.


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