Difference between revisions of "Team:Amsterdam/Modeling"

 
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           <h2>Modelling and simulations</h2>
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           <h2>Modelling</h2>
 
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             <h4>Trying to find a match</h4>
 
             <h4>Trying to find a match</h4>
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                               <h3>Background</h3>
 
                               <h3>Background</h3>
 
                              
 
                              
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                             <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.  
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                             <p>In our project we focussed on the <a href="https://2015.igem.org/Team:Amsterdam/Project/Stability">stability </a> in different ways. On a mathematical point of view there are also different definitions on stability. A set of differential equations can have stable steady states for example, but you could also argue that convergence to certain solutions is a measure for robustness and stability. Here we will also shine a light on these types of stability. These simulations influence the way we envision further applications.
 
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                               <h3>Aim</h3>
 
                               <h3>Aim</h3>
 
                              
 
                              
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                             <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).
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                             <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>
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                               <h3>Results</h3>
 
                               <h3>Results</h3>
 
                              
 
                              
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                            <ul style = "font-family: 'Montserrat', sans-serif">
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<p>
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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.
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We analyzed the ratio of biomass of the different cultures. We can see that in a lot of cases this ratio converges. Important parameters for what ratio the plots converge to seem to be the mus of the different species. The initial conditions of the ratios when grown in a batch seem not that important, as is also shown in the lab. The ratio we arrive at in the lab however, seem different. This seems to hint at wrongly chosen or inaccurately measured parameters, however we can still study the long term behavior.
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                                         <h3>Connections</h3>
 
                                         <h3>Connections</h3>
 
                                     <p>
 
                                     <p>
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                     <p>To create models as accurate as possible, we measured these in wet lab. Here the results of the physiology</p>
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                                          <h3></h3>
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                     <p>Sometimes modellers tend to be the lone wolfs in a project. We didn't want this to happen, so there are some clear connections between the tools we created with modelling and the wet lab. Initially the need to search for compounds which could be produced genetically stable, came from the wet lab, where we saw that most producing strains are <a href="https://2015.igem.org/Team:Amsterdam/Project/Phy_param/Synechocysytis">unstable</a>. Before we even started <a href="https://2015.igem.org/Team:Amsterdam/Project/Eng_rom/Photosyn_car"> engineering <i>Synechocystis</i></a>, we wanted to find out whether we could produce a compound genetically <a href="https://2015.igem.org/Team:Amsterdam/Project/Stability">stable</a>. This is where the Stable Compound Generater comes in. We also needed to <a href="https://2015.igem.org/Team:Amsterdam/Project/Eng_rom/Dependecies">engineer an auxotroph</a> in order to to use serial propagations of consortia in <a href="https://2015.igem.org/Team:Amsterdam/Project/emulsions">emulsions</a> to find a more robust consortium. Both algorithms provided information which was really used in the lab.</p>
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                                     </p>
 
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  <figure class ="image fit" style = "align:center">
 
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   <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
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   <img src="https://static.igem.org/mediawiki/2015/c/c0/Amsterdam_eq1.png" alt="ODE">
 
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  <figure class ="image fit" style = "align:center">
 
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   <img src="https://2015.igem.org/File:Amsterdam_monodeq.png" alt="Monod equation">
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   <img src="https://static.igem.org/mediawiki/2015/c/c6/Amsterdam_monodeq.png" alt="Monod equation">
 
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   <img src="https://2015.igem.org/File:Amsterdam_eq1.png" alt="ODE">
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   <img src="https://static.igem.org/mediawiki/2015/c/c0/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>  
 
</figure>  
 
<p>
 
<p>
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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:
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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:
  
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                                    <a href="drylab.htmldrylab_kinetic">
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<br>
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<h4>Reliance on each other</h4>
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<figure class ="image fit" style = "align:center">
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                                    </a>
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  <img src="https://static.igem.org/mediawiki/2015/9/9d/Amsterdam_independent_batch.png" alt="ODE">
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</figure>  
 
<p>
 
<p>
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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.
 
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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.
 
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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/y<sub>syn/s</sub>.
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            <section id="algorithms_approach" class="wrapper style4">
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                <h3>Approach</h3>
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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
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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>
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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.
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                                <section>
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                                    <a href="drylab.html#drylab_kinetic">
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<h4>Stable Compound Generator</h4>
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                                    </a>
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<p>
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In previous section we emphasized we wanted to create <SPAN  CLASS="textbf">stable</SPAN> carbon compound producing cyanobacterial strain.
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But how do we do this? We created an algorithm which finds ways to make a stable producer.  
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It is based on two ideas:
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<OL>
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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,
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<LI>It takes a lot more evolutionary time to re-create a whole new gene than to accumulate a loss-of-function mutation.
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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
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</LI>
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<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.
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</LI>
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</OL>
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Below follows a general outline of the algorithm.
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<BR>The algorithm makes a list of carbon compounds associated with the production of biomass in the cell.
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<BR>For each of these compounds, the algorithm does the following:
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<BR>
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<UL>
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<LI>Find sources reactions of the compound in the extracellular space and set the boundaries of these reactions to zero.
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</LI>
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<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
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</LI>
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<LI>Find the genes associated to the reactions.
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</LI>
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<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
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</LI>
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<LI>Make a list of combinations of the genes associated to the production of a compound.
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</LI>
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<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.
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</LI>
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<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.
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</LI>
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<LI>Do a flux balance analysis on the model en check if biomass is formed.
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</LI>
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<LI>Check the value of the sink of the compound of interest.  
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</LI>
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</UL>
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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.
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<figure class ="image fit" style = "align:center">
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  <img src="https://static.igem.org/mediawiki/2015/d/d0/Min.png" alt="growth rate dependencies">
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</figure>
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<p>
  
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                            <div class="6u">
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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.
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                                <section>
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                                    <a href="drylab.htmldrylab_kinetic">
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</p>
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<h4>Auxotrpophy Sniper</h4>
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</figure>
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                                    </a>
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<div class="8u">
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<figure class ="image fit" style = "align:center">
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  <img src="https://2015.igem.org/File:Interdependent_substrate.png" alt="ODE">
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</figure>  
 
<p>
 
<p>
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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.
 
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Here follows an overview of how the Auxotrophy Sniper works:
 
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<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.
 
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For each metabolite on the list it does the following:
 
  
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<UL>
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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
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<LI>Set flux boundaries of source reaction in the extracellular space (if there is already one present) to zero.
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produce in a growth coupled way, but has a constant production per amount of biomass.
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</LI>
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<LI>Find all primary reactions associated to the compound we want to make the organism dependent on.
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<h4>Turbidostat</h4>
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</LI>
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<LI>For each primary reaction, find genes which are associated to the primary reactions.
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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,
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</LI>
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but the amount of biomass must still become constant. This means the following:
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<LI>For each gene find all reactions which have the gene in their gene association (secondary reactions).
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</LI>
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<LI>Make a list of possible combinations of genes which can be knocked out.
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</LI>
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<LI>Go over the combinations one by one, for each gene in a combination turn off primary and secondary reactions.
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</LI>
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<LI>Per combination, do a flux balance analysis and check for biomass formation (growth).
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</LI>
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<LI>If it does <SPAN  CLASS="textbf">not</SPAN> form any biomass, a source reaction of the compound is added to the model
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</LI>
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<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.
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</LI>
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</UL>
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<P>
 
  
 
</p>
 
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  <img src="https://2015.igem.org/File:Amsterdam_turbidoexamplefg.png" alt="ODE">
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                <header class="major">
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                <h3>results</h3>
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<p>
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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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                                <section>
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In a chemostat it would mean that D < μ is a chosen dilution rate and that μ becomes equal to D due to
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substrate limitation. In a turbidostat however, D becomes equal to μ.
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In the case where there are two strains, strain a and b, that share a turbidostat, the differential equations that
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then describe the system looks like the following:
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</p>
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</div>
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</div>
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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_turbioexample3.png" alt="ODE">
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</figure>  
 
<p>
 
<p>
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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 
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Then the following holds:
 
</p>
 
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For the turbidostat we then arrive at the following set of differential equations:
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  <img src="https://2015.igem.org/File:Amsterdam_turbido_final_dif.png" alt="ODE">
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where:
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https://2015.igem.org/File:Amsterdam_helper.png
 
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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.
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Latest revision as of 03:59, 19 September 2015

iGEM Amsterdam 2015

Modelling

Trying to find a match

Overview

Background

In our project we focussed on the stability in different ways. On a mathematical point of view there are also different definitions on stability. A set of differential equations can have stable steady states for example, but you could also argue that convergence to certain solutions is a measure for robustness and stability. Here we will also shine a light on these types of stability. These simulations influence the way we envision further applications.

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 analyzed the ratio of biomass of the different cultures. We can see that in a lot of cases this ratio converges. Important parameters for what ratio the plots converge to seem to be the mus of the different species. The initial conditions of the ratios when grown in a batch seem not that important, as is also shown in the lab. The ratio we arrive at in the lab however, seem different. This seems to hint at wrongly chosen or inaccurately measured parameters, however we can still study the long term behavior.

Connections

Sometimes modellers tend to be the lone wolfs in a project. We didn't want this to happen, so there are some clear connections between the tools we created with modelling and the wet lab. Initially the need to search for compounds which could be produced genetically stable, came from the wet lab, where we saw that most producing strains are unstable. Before we even started engineering Synechocystis, we wanted to find out whether we could produce a compound genetically stable. This is where the Stable Compound Generater comes in. We also needed to engineer an auxotroph in order to to use serial propagations of consortia in emulsions to find a more robust consortium. Both algorithms provided information which was really used in the lab.

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

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:


ODE

Herein is syn the amount of biomass of Synechocystis and ec the amount of biomass of E. coli. ys is the substrate yield of Synechocystis. Since in this model the substrate is only formed when Synechocystis 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 E. coli has a much higher growth rate. In this model, Synechocystis is not dependent on E. coli. There are two ways in which Synechocystis may be dependent on E. coli, which we have explored. Firstly, E. coli may produce a substrate Synechocystis grows on, as is the case with the auxotrophic Synechocystis. Secondly, E. coli may decrease the light intensity in the culture, in this way slowing down the growth of Synechocystis. The growth rate of Synechocystis 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 Synechocystis would be