Difference between revisions of "Team:Amsterdam/Modeling"

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{{Amsterdam}}
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{{Amsterdam/navbar}}
 
<html>
 
<html>
 +
        <!-- header -->
 +
    <section id="modelling_header" class="wrapper style8">
 +
        <header class="major">
 +
          <h2>Modelling and simulations</h2>
 +
<p>
 +
            <h4>Trying to find a match</h4>
 +
</p>
 +
        </header>
 +
    </section>
  
−
<h2> Modeling</h2>
 
  
  
−
<div class="highlightBox">
+
    <!-- Overview -->
−
<h4>Note</h4>
+
<section id="algorithms_header" class="wrapper style8">
−
<p>In order to be considered for the <a href="https://2015.igem.org/Judging/Awards#SpecialPrizes">Best Model award</a>, you must fill out this page.</p>
+
<header class="major">
−
</div>
+
<h3>Overview</h3>
 +
</header>
 +
<div class="container">
 +
<div class="row">                                                
 +
                        <div class="3u">
  
 +
                        <section>
 +
                              <h3>Background</h3>
 +
                           
 +
                            <p>In our project we focussed on the <a href="https://2015.igem.org/Team:Amsterdam/Project/Stability">stability </a> in different ways. In mathematical sense there are also different ways of stability. Steady states in differential equations may be stable or not. But a set of differential equations may also converge to an interesting solution. This convergence may also be called stability.
 +
</p>
 +
                              <h3>Aim</h3>
 +
                           
 +
                            <p>We want to answer questions about the dynamics of an engineered consortium, which will help in envisioning an archetypal final application. What will happen to the growth rate of the organisms during the cultivation? With what initial conditions and parameters will the ratio between the biomass of the different species converge to the same ratio? And if so, what will that ratio be? What is the influence of the light intensity on Synechocystis and how will this be influenced by the presence of a chemoheterotroph which blocks and scatters light? To answer these questions we created kinetic models consisting of ordinary differential equations (ODEs).
 +
</p>
  
−
<p>Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.</p>
+
</section>              
 +
                        </div>
  
−
<p>
+
 
−
Here are a few examples from previous teams:
+
                        <div class="3u">
 +
                        <section>
 +
                              <h3>Approach</h3>
 +
 
 +
                            <p>We modelled the biomass per liter of Synechocystis and E. coli in chemostat (being the turbidostat a particular case of the chemostat in which the D = umax) as well as batch cultures. We created a set of ODEs and analyzed them with pydstool - a python tool, which can solve differential equations numerically. Model parameters were measured as accurate as possible in the wet lab specifically for this purpose, i.e. in silico simulations that increase and test our understanding of the underlying interactions.
 
</p>
 
</p>
−
<ul>
+
</section>
−
<li><a href="https://2014.igem.org/Team:ETH_Zurich/modeling/overview">ETH Zurich 2014</a></li>
+
</div>
−
<li><a href="https://2014.igem.org/Team:Waterloo/Math_Book">Waterloo 2014</a></li>
+
<div class="3u">
 +
<section>
 +
                              <h3>Results</h3>
 +
                           
 +
                            <ul style = "font-family: 'Montserrat', sans-serif">
 +
We show that consortia may show a robustness to the initial conditions. We looked at behaviour in different environments. We show simulations of these models.
 
</ul>
 
</ul>
 +
</p>
 +
</section>   
 +
                                       
 +
                        </div>
 +
                            <div class="3u">
 +
                                <section>
 +
                                        <h3>Connections</h3>
 +
                                    <p>
 +
                    <p>To create models as accurate as possible, we measured these in wet lab. Here the results of the physiology</p>
 +
                                    </p>
 +
                                </section>
 +
                            </div> <!-- 6u -->
 +
               
 +
                        </div>
 +
                        </div>
 +
</div>
  
  
 +
 +
 +
<!-- questions and methods -->
 +
            <section id="algorithms_aim" class="wrapper style2">
 +
                <header class="major">
 +
                <h3>approach</h3>
 +
   
 +
                </header>
 +
                    <div class="container">
 +
                        <div class="row">
 +
 +
                            <div class="8u">
 +
                                <section>
 +
 +
<h4>Batch</h4>
 +
<p>
 +
Unlimited cell growth is exponential. The amount of biomass per time for an exponentially growing species can be given by the following ODE:
 +
</p>
 +
</figure>
 
</div>
 
</div>
 +
</div>
 +
<div class="8u">
 +
<br>
 +
<figure class ="image fit" style = "align:center">
 +
  <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
 +
</figure>
 +
<p>
 +
Herein is a the amount of biomass per liter and &mu; the growth rate normalized for biomass. One can easily verify that the solution of this differential equation is indeed exponential growth (a = ce<sup>&mu;t</sup> ). Now from experimental data it has been shown that limited growth on a substrate is a bit different. The normalized growth rate is then dependent on the concentration of substrate. According to the monod equation, the &mu; is dependent on [S] in
 +
the following way:
 +
</p>
 +
</figure>
 +
</div>
 +
</div>
 +
<div class="8u">
 +
<br>
 +
<figure class ="image fit" style = "align:center">
 +
  <img src="https://2015.igem.org/File:Amsterdam_monodeq.png" alt="Monod equation">
 +
</figure>
 +
<p>
 +
Herein is &mu;<sub>max</sub> the maximal growth rate (equal to the growth rate at unlimited growth), and [S] the concentration of substrate. kS is the concentration of [S] at a rate 1/2 &mu;<sub>max</sub>. We also know this equation from enzyme kinetics as the Michaelis-Menten equation. In enzyme kinetics this equation is used to calculate the rates at which enzymes convert products. However the Michaelis-Menten equation is based on theoretical arguments, while Monod is based on experimental findings. According to Monod, the growth rate saturates as the concentration becomes higher (see figure 1).
 +
</p>
 +
</figure>
 +
</div>
 +
</div>
 +
<div class="8u">
 +
<br>
 +
<figure class ="image fit" style = "align:center">
 +
  <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
 +
  <figcaption>Figure 1: Limited growth on a substrate according to the Monod equation. &mu; is the normalized growth rate in units per hour &mu;<sub>max</sub> is the maximal growth rate and [S] is the substrate  concentration. kS is the concentration
 +
at the rate equal to 1/2 &mu;<sub>max</sub>.</figcaption>
 +
</figure>
 +
<p>
 +
In our consortium, E. coli grows limited on acetate. So now we know how μ depends on the concentration of acetate. We now need to model the concentration of acetate. We assume the synthesis of acetate is growth coupled and depends linearly on the growth of Synechocystis. We also know that the maximal uptake rate of acetate by E. coli is dependent 1/y herein is y the yield of E. coli on acetate in is in milligram Dry Weight E. coli per millimole acetate per liter. The uptake of substrate per unit time is also in a saturable way dependent on the concentration of S, exactly the same way the growth speed is dependent on the the concentration of substrate. The synthesis of the substrate is growth coupled and is dependent on the amount of biomass of Synechocystis formed in time. We arrive at the following differential equations:
 +
 +
                                </section>
 +
                               
 +
                            </div> <!-- 6u -->
 +
</div>
 +
                            <div class="6u">
 +
                                <section>
 +
                                    <a href="drylab.htmldrylab_kinetic">
 +
<h4>Reliance on each other</h4>
 +
                                    </a>
 +
<p>
 +
Auxotrophs are organisms which need a certain compound which they can take up from their environment to be able to grow. They can be very useful in synthetic biology, as an auxotrophy can regulate the growth of a certain organism. It is also useful synthetic consortium, as it can create a dependency of one species on another. In our project we wanted to make the chemoheterotroph dependent on the cyanobacterium by the carbon compound the cyanobacteria produces, because this will be part of the flux of CO<sub>2</sub> into product. However, we also wanted to create dependency of the cyanobacterium on the chemoheterotroph, because it can stabilize a consortium if, for example, the chemoheterotoph has a lower growth rate than the photoautotroph on the carbon compound. Other important reasons are that this dependency will decrease the risk of the cyanobacterium surviving on its own in the environment in case of an outbreak. Some chemoheterotrophs are very efficient in the production of a certain compound. If the photoautotroph does not have to synthesize the compound itself, it may increase the growth rate of the photoautotroph. This increases the rate at which carbon is fixated and carbon compound is produced, which in its turn benefits the chemoheterotroph. It is also suggested that this interdependency may create a more robust system.
 +
So the second question that we had in the lab is how can we create an auxotroph? To answer this question we also created an algorithm, the Auxotrophy Sniper.
 +
</p>
 +
                                </section>
 +
                               
 +
                            </div> <!-- 6u -->
 +
                        </div> <!-- row -->
 +
                    </div> <!-- container -->
 +
            </section>
 +
 +
<!-- questions and methods -->
 +
            <section id="algorithms_approach" class="wrapper style4">
 +
                <header class="major">
 +
                <h3>Approach</h3>
 +
                </header>
 +
    <div class="6u">
 +
    <div class = "container">
 +
      <div class="row">
 +
    <p>
 +
For some questions we had (see Aim) we decided to look for an answer based on algorithms for genome scale flux balance analysis (FBA) models. We have used genome scale FBA that already existed of <it>Synechocystis</it> and PySCeS CBMpy, a tool for <b>C</b>onstraint <b>B</b>ased <b>M</b>odeling using <b>Py</b>thon <b>S</b>imulator for <b>Ce</b>llular <b>S</b>ystems (<a
 +
href="http://pysces.sourceforge.net/">Olivier, 2014-2015</a>). In flux balance analysis (FBA) you assume that all reactions in a cell are in steady state and then you can then represent them in a set of linear equations. The objective function then optimizes something given a set of boundary conditions about the flux through each reaction. The objective function is often the formation of biomass. If a cell is producing biomass, means it grows.</p>
 +
                    <div class="container">
 +
                        <div class="row">
 +
 +
                            <div class="6u">
 +
                                <section>
 +
                                    <a href="drylab.html#drylab_kinetic">
 +
<h4>Stable Compound Generator</h4>
 +
                                    </a>
 +
<p>
 +
In previous section we emphasized we wanted to create <SPAN  CLASS="textbf">stable</SPAN> carbon compound producing cyanobacterial strain.
 +
But how do we do this? We created an algorithm which finds ways to make a stable producer.
 +
It is based on two ideas:
 +
 +
<OL>
 +
<LI>It takes a lot more evolutionary time to re-create a whole new gene than to accumulate a loss-of-function mutation.
 +
</LI>
 +
<LI>If the gene responsible for the production of the compound is expressed only when the organism grows (growth coupled production), the organism cannot simply stop expressing the pathway.
 +
</LI>
 +
</OL>
 +
Below follows a general outline of the algorithm.
 +
<BR>The algorithm makes a list of carbon compounds associated with the production of biomass in the cell.
 +
<BR>For each of these compounds, the algorithm does the following:
 +
<BR>
 +
<UL>
 +
<LI>Find sources reactions of the compound in the extracellular space and set the boundaries of these reactions to zero.
 +
</LI>
 +
<LI>Find all reactions associated with the compound, as the model only contains reactions with gene associations and no genes which directly influence flux. We will call these reactions primary reactions
 +
</LI>
 +
<LI>Find the genes associated to the reactions.
 +
</LI>
 +
<LI>For each of these genes, find all reactions which have the gene of interest in their gene association. We will call these reactions secondary reactions
 +
</LI>
 +
<LI>Make a list of combinations of the genes associated to the production of a compound.
 +
</LI>
 +
<LI>For each of the genes involved each combination, set flux boundaries of these reactions to zero. In that way we simulate the knock-out of the gene.
 +
</LI>
 +
<LI>Since accumulation of compound is not possible in FBA, while in reality most cells are leaking, we create sinks for compounds which would otherwise possibly accumulate. These sinks prevent the model from going to a non-growing steady state, because the compound would otherwise accumulate.
 +
</LI>
 +
<LI>Do a flux balance analysis on the model en check if biomass is formed.
 +
</LI>
 +
<LI>Check the value of the sink of the compound of interest.
 +
</LI>
 +
</UL>
 +
If there is still formation of biomass (growth) and the sink of the compound of interest is used to export the compound out of the cell, the knock-out of the combination of genes is a good candidate for making a stable producer.
 +
 +
<P>
 +
</p>
 +
                                </section>
 +
                               
 +
                            </div> <!-- 6u -->
 +
 +
                            <div class="6u">
 +
                                <section>
 +
                                    <a href="drylab.htmldrylab_kinetic">
 +
<h4>Auxotrpophy Sniper</h4>
 +
                                    </a>
 +
<p>
 +
We wanted to create an auxotrophic <SPAN  CLASS="textit">Synechocystis</SPAN> strain as part of our consortium. But how can we make an auxotroph out of <SPAN  CLASS="textit">Synechocystis</SPAN>? To get the answer to this question we decided to create an algorithm that works on genome scale FBA models. The general idea is that an auxotroph can be created by knocking out a combination of genes involved in the production of the compound we want to make a dependency on. The organism should not be able to grow if there if these genes are knocked out and there is no source for the compound in the extracellular space. When this is the case we create a source in the extracellular space (add compound to the medium) and after that the organism should be able to grow again.
 +
Here follows an overview of how the Auxotrophy Sniper works:
 +
<BR>The algorithm takes a list of compounds of which you want to make a synthesis deficiency in the organism of choice. This list can contain for example vitamins, or amino acids.
 +
For each metabolite on the list it does the following:
 +
 +
<UL>
 +
<LI>Set flux boundaries of source reaction in the extracellular space (if there is already one present) to zero.
 +
</LI>
 +
<LI>Find all primary reactions associated to the compound we want to make the organism dependent on.
 +
</LI>
 +
<LI>For each primary reaction, find genes which are associated to the primary reactions.
 +
</LI>
 +
<LI>For each gene find all reactions which have the gene in their gene association (secondary reactions).
 +
</LI>
 +
<LI>Make a list of possible combinations of genes which can be knocked out.
 +
</LI>
 +
<LI>Go over the combinations one by one, for each gene in a combination turn off primary and secondary reactions.
 +
</LI>
 +
<LI>Per combination, do a flux balance analysis and check for biomass formation (growth).
 +
</LI>
 +
<LI>If it does <SPAN  CLASS="textbf">not</SPAN> form any biomass, a source reaction of the compound is added to the model
 +
</LI>
 +
<LI>Another flux balance analysis of the model is done and if there is biomass formation now, the combination of genes are good candidates. If this combination of genes is knocked out the organism will probably become an auxotroph.
 +
</LI>
 +
</UL>
 +
 +
<P>
 +
 +
</p>
 +
                                </section>
 +
                               
 +
                            </div> <!-- 6u -->
 +
                        </div> <!-- row -->
 +
                    </div> <!-- container -->
 +
            </section>
 +
            <section id="algorithms_aim" class="wrapper style8">
 +
                <header class="major">
 +
                <h3>results</h3>
 +
   
 +
                </header>
 +
                    <div class="container">
 +
                        <div class="row">
 +
 +
                            <div class="8u">
 +
                                <section>
 +
<p>
 +
We set out to create two algorithms. One to find ways to create an organism which can produce a carbon compound in a genetic way, and another organisms which can 
 +
</p>
 +
                                </section>
 +
                               
 +
                         
 +
                    </div> <!-- container -->
 +
            </section>
  
 
</html>
 
</html>
 +
 +
 +
 +
</p>
 +
</figure>
 +
</div>
 +
</div>
 +
<div class="8u">
 +
<br>
 +
<figure class ="image fit" style = "align:center">
 +
  <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
 +
</figure>
 +
<p>

Revision as of 02:47, 19 September 2015

iGEM Amsterdam 2015

Modelling and simulations

Trying to find a match

Overview

Background

In our project we focussed on the stability in different ways. In mathematical sense there are also different ways of stability. Steady states in differential equations may be stable or not. But a set of differential equations may also converge to an interesting solution. This convergence may also be called stability.

Aim

We want to answer questions about the dynamics of an engineered consortium, which will help in envisioning an archetypal final application. What will happen to the growth rate of the organisms during the cultivation? With what initial conditions and parameters will the ratio between the biomass of the different species converge to the same ratio? And if so, what will that ratio be? What is the influence of the light intensity on Synechocystis and how will this be influenced by the presence of a chemoheterotroph which blocks and scatters light? To answer these questions we created kinetic models consisting of ordinary differential equations (ODEs).

Approach

We modelled the biomass per liter of Synechocystis and E. coli in chemostat (being the turbidostat a particular case of the chemostat in which the D = umax) as well as batch cultures. We created a set of ODEs and analyzed them with pydstool - a python tool, which can solve differential equations numerically. Model parameters were measured as accurate as possible in the wet lab specifically for this purpose, i.e. in silico simulations that increase and test our understanding of the underlying interactions.

Results

    We show that consortia may show a robustness to the initial conditions. We looked at behaviour in different environments. We show simulations of these models.

Connections

To create models as accurate as possible, we measured these in wet lab. Here the results of the physiology

approach

Batch

Unlimited cell growth is exponential. The amount of biomass per time for an exponentially growing species can be given by the following ODE:


ODE

Herein is a the amount of biomass per liter and μ the growth rate normalized for biomass. One can easily verify that the solution of this differential equation is indeed exponential growth (a = ceμt ). Now from experimental data it has been shown that limited growth on a substrate is a bit different. The normalized growth rate is then dependent on the concentration of substrate. According to the monod equation, the μ is dependent on [S] in the following way:


Monod equation

Herein is μmax the maximal growth rate (equal to the growth rate at unlimited growth), and [S] the concentration of substrate. kS is the concentration of [S] at a rate 1/2 μmax. We also know this equation from enzyme kinetics as the Michaelis-Menten equation. In enzyme kinetics this equation is used to calculate the rates at which enzymes convert products. However the Michaelis-Menten equation is based on theoretical arguments, while Monod is based on experimental findings. According to Monod, the growth rate saturates as the concentration becomes higher (see figure 1).


ODE
Figure 1: Limited growth on a substrate according to the Monod equation. μ is the normalized growth rate in units per hour μmax is the maximal growth rate and [S] is the substrate concentration. kS is the concentration at the rate equal to 1/2 μmax.

In our consortium, E. coli grows limited on acetate. So now we know how μ depends on the concentration of acetate. We now need to model the concentration of acetate. We assume the synthesis of acetate is growth coupled and depends linearly on the growth of Synechocystis. We also know that the maximal uptake rate of acetate by E. coli is dependent 1/y herein is y the yield of E. coli on acetate in is in milligram Dry Weight E. coli per millimole acetate per liter. The uptake of substrate per unit time is also in a saturable way dependent on the concentration of S, exactly the same way the growth speed is dependent on the the concentration of substrate. The synthesis of the substrate is growth coupled and is dependent on the amount of biomass of Synechocystis formed in time. We arrive at the following differential equations:

Reliance on each other

Auxotrophs are organisms which need a certain compound which they can take up from their environment to be able to grow. They can be very useful in synthetic biology, as an auxotrophy can regulate the growth of a certain organism. It is also useful synthetic consortium, as it can create a dependency of one species on another. In our project we wanted to make the chemoheterotroph dependent on the cyanobacterium by the carbon compound the cyanobacteria produces, because this will be part of the flux of CO2 into product. However, we also wanted to create dependency of the cyanobacterium on the chemoheterotroph, because it can stabilize a consortium if, for example, the chemoheterotoph has a lower growth rate than the photoautotroph on the carbon compound. Other important reasons are that this dependency will decrease the risk of the cyanobacterium surviving on its own in the environment in case of an outbreak. Some chemoheterotrophs are very efficient in the production of a certain compound. If the photoautotroph does not have to synthesize the compound itself, it may increase the growth rate of the photoautotroph. This increases the rate at which carbon is fixated and carbon compound is produced, which in its turn benefits the chemoheterotroph. It is also suggested that this interdependency may create a more robust system. So the second question that we had in the lab is how can we create an auxotroph? To answer this question we also created an algorithm, the Auxotrophy Sniper.