Difference between revisions of "Team:Amsterdam/Modeling"

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  <figure class ="image fit" style = "align:center">
 
  <figure class ="image fit" style = "align:center">
 
   <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
 
   <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
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  <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
 
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at the rate equal to 1/2 &mu;<sub>max</sub>.</figcaption>
 
 
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</figure>  
 
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<p>
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(link turbidostat) If we want to model the consortium in a turbidostat, we have to account for the fact that both <it>Synechocystis</it> and <it>E. coli</it> are increasing the OD as they grow. This means that the dilution rate is dependent on the biomass of <it>Synechocystis</it> as well as that of <it>E. coli</it>. For simplicity we make the assumption that there is a constant flow through the system, instead of only diluting when the threshold is reached. To understand this we first look at the case of a single strain, called b. In a chemostat the growth rate of the organism would become equal to the dilution rate. In a turbidostat however, an organism can grow at its maximal growth rate,
 
(link turbidostat) If we want to model the consortium in a turbidostat, we have to account for the fact that both <it>Synechocystis</it> and <it>E. coli</it> are increasing the OD as they grow. This means that the dilution rate is dependent on the biomass of <it>Synechocystis</it> as well as that of <it>E. coli</it>. For simplicity we make the assumption that there is a constant flow through the system, instead of only diluting when the threshold is reached. To understand this we first look at the case of a single strain, called b. In a chemostat the growth rate of the organism would become equal to the dilution rate. In a turbidostat however, an organism can grow at its maximal growth rate,
 
but the amount of biomass must still become constant. This means the following:
 
but the amount of biomass must still become constant. This means the following:
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<div class="8u">
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<br>
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Amsterdam_turbidoexamplefg.png" alt="ODE">
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</figure>
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<p>
  
 
Where b is the amount of biomass of the strain b, mu is the growth rate and D the dilution rate.
 
Where b is the amount of biomass of the strain b, mu is the growth rate and D the dilution rate.
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In the case where there are two strains, strain a and b, that share a turbidostat, the differential equations that
 
In the case where there are two strains, strain a and b, that share a turbidostat, the differential equations that
 
then describe the system looks like the following:
 
then describe the system looks like the following:
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<br>
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Amsterdam_turbioexample3.png" alt="ODE">
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</figure>
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<p>
  
 
Then the following holds:
 
Then the following holds:
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<div class="8u">
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<br>
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Amsterdam_turbidologic.png" alt="ODE">
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</figure>
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<p>
  
 
For the turbidostat we then arrive at the following set of differential equations:
 
For the turbidostat we then arrive at the following set of differential equations:
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</figure>
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<div class="8u">
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<br>
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Amsterdam_turbido_final_dif.png" alt="ODE">
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</figure>
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<p>
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where:
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https://2015.igem.org/File:Amsterdam_helper.png
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Amsterdam_helper.png" alt="ODE">
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<p>
  
 
Or if we take again the light limited model:
 
Or if we take again the light limited model:

Revision as of 03:37, 19 September 2015

</p> </section> 

                       </div>	

                               <section>

Connections

<p>To create models as accurate as possible, we measured these in wet lab. <a href="https://2015.igem.org/Team:Amsterdam/Project/Phy_param">Here the results of the physiology measurements<a> of <it>E. coli</it> and <it>Synechocystis</it> 

                                   </p>
                               </section>

                       </div>
                       </div>

</div> 



           <section id="algorithms_aim" class="wrapper style2">
               <header class="major">

approach

               </header>

                               <section>

Batch

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

</figure> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">

</figure>

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:

</figure> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_monodeq.png" alt="Monod equation">

</figure>

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).

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">

</figure>

In our consortium, <it>E. coli</it> 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 <it>Synechocystis</it>. We also know that the maximal uptake rate of acetate by <it>E. coli</it> is dependent 1/y herein is y the yield of <it>E. coli</it> on acetate in is in milligram Dry Weight <it>E. coli</it> 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 <it>Synechocystis</it> formed in time. We arrive at the following differential equations:

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_independent_batch.png" alt="ODE">

</figure>

Herein is syn the amount of biomass of <it>Synechocystis</it> and ec the amount of biomass of <it>E. coli</it>. ys is the substrate yield of <it>Synechocystis</it>. Since in this model the substrate is only formed when <it>Synechocystis</it> forms biomass, there is a constant amount of substrate formed. The yield is usually expressed in 5 gram dry weight per mole substrate used. In this case we mean gram dry weight mole substrate per formed. So to find the amount of substrate that is formed per amount of biomass that is formed we simply take 1/ysyn/s. It can be easily seen that such a relationship will not be stable if μmax,ec << μmax,cyn, but in most cases <it>E. coli</it> has a much higher growth rate. In this model, <it>Synechocystis</it> is not dependent on <it>E. coli</it>. There are two ways in which <it>Synechocystis</it> may be dependent on <it>E. coli</it>, which we have explored. Firstly, <it>E. coli</it> may produce a substrate <it>Synechocystis</it> grows on, as is the case with the auxotrophic <it>Synechocystis</it>. Secondly, <it>E. coli</it> may decrease the light intensity in the culture, in this way slowing down the growth of <it>Synechocystis</it>. The growth rate of <it>Synechocystis</it> can only be limited by one of these two processes and it will always be limited by the process that slows it the most. Either μsyn is lower than μmax,syn because there is a photon shortage, but then the amount of substrate available at that growth rate would be enough, or the amount of substrate is limiting, but then the amount of photons available would also be enough for that given growth rate. So actually the growth rate of <it>Synechocystis</it> would be

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Min.png" alt="growth rate dependencies">

</figure>

Herein is f (syn, ec) a function which determines the factor of decrease in growth rate because of a photon shortage and it is a function of the amount of biomass per liter of <it>Synechocystis</it> as well as that of <it>E. coli</it>. If we now assume that the amount of substrate <it>E. coli</it> produces is going to be limiting we arrive at the following set of ODEs.

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Interdependent_substrate.png" alt="ODE">

</figure>

Herein is Qp,ec the amount of [S2] formed by <it>E. coli</it> per gram dry weight of <it>E. coli</it>. We assume <it>E. coli</it> does not produce in a growth coupled way, but has a constant production per amount of biomass.

Turbidostat

(link turbidostat) If we want to model the consortium in a turbidostat, we have to account for the fact that both <it>Synechocystis</it> and <it>E. coli</it> are increasing the OD as they grow. This means that the dilution rate is dependent on the biomass of <it>Synechocystis</it> as well as that of <it>E. coli</it>. For simplicity we make the assumption that there is a constant flow through the system, instead of only diluting when the threshold is reached. To understand this we first look at the case of a single strain, called b. In a chemostat the growth rate of the organism would become equal to the dilution rate. In a turbidostat however, an organism can grow at its maximal growth rate, but the amount of biomass must still become constant. This means the following:


</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_turbidoexamplefg.png" alt="ODE">

</figure>

Where b is the amount of biomass of the strain b, mu is the growth rate and D the dilution rate. In a chemostat it would mean that D < μ is a chosen dilution rate and that μ becomes equal to D due to substrate limitation. In a turbidostat however, D becomes equal to μ. In the case where there are two strains, strain a and b, that share a turbidostat, the differential equations that then describe the system looks like the following:

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_turbioexample3.png" alt="ODE">

</figure>

Then the following holds:

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_turbidologic.png" alt="ODE">

</figure>

For the turbidostat we then arrive at the following set of differential equations:

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_turbido_final_dif.png" alt="ODE">

</figure>

where: https://2015.igem.org/File:Amsterdam_helper.png

</figure>

</div> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_helper.png" alt="ODE">

</figure>

Or if we take again the light limited model: Models are very nice, but if you cannot verify them, they are a fantasy. We tried to measure the different parameters in the models as accurate as possible. The methods and data of these measurements for <it>Synechocystis</it> can be found here {link}. For <it>E. coli</it>, they can be found here{link}. </section> </div> </div> </div> 

                               <section>

                               <section>

Turbidostat

<p> </p> 

                               </section>
                               

           </section>

           <section id="algorithms_approach" class="wrapper style4">
               <header class="major">

Approach

               </header>

   <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 Constraint Based Modeling using Python Simulator for Cellular Systems (<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>

                               <section>
                                   <a href="drylab.html#drylab_kinetic">

Stable Compound Generator

                                   </a>

<p> In previous section we emphasized we wanted to create stable 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:

  1. It takes a lot more evolutionary time to re-create a whole new gene than to accumulate a loss-of-function mutation.
  2. 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.

Below follows a general outline of the algorithm.
The algorithm makes a list of carbon compounds associated with the production of biomass in the cell.
For each of these compounds, the algorithm does the following:

  • Find sources reactions of the compound in the extracellular space and set the boundaries of these reactions to zero.
  • 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
  • Find the genes associated to the reactions.
  • 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
  • Make a list of combinations of the genes associated to the production of a compound.
  • 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.
  • 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.
  • Do a flux balance analysis on the model en check if biomass is formed.
  • Check the value of the sink of the compound of interest.

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>
                               

                               <section>
                                   <a href="drylab.htmldrylab_kinetic">

Auxotrpophy Sniper

                                   </a>

<p> We wanted to create an auxotrophic Synechocystis strain as part of our consortium. But how can we make an auxotroph out of Synechocystis? 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:
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:

  • Set flux boundaries of source reaction in the extracellular space (if there is already one present) to zero.
  • Find all primary reactions associated to the compound we want to make the organism dependent on.
  • For each primary reaction, find genes which are associated to the primary reactions.
  • For each gene find all reactions which have the gene in their gene association (secondary reactions).
  • Make a list of possible combinations of genes which can be knocked out.
  • Go over the combinations one by one, for each gene in a combination turn off primary and secondary reactions.
  • Per combination, do a flux balance analysis and check for biomass formation (growth).
  • If it does not form any biomass, a source reaction of the compound is added to the model
  • 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.

<P>

</p> 

                               </section>
                               

           </section>
           <section id="algorithms_aim" class="wrapper style8">
               <header class="major">

results

               </header>

                               <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>
                               
                          

           </section>

</html>



</p> </figure> 


<figure class ="image fit" style = "align:center">
 <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">

</figure>

<p>