Team:Braunschweig/Modeling-content
From 2014.igem.org
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Our mathematical model, based on laboratory and literature data, provides an overview of the efficiency and impact of our system. | Our mathematical model, based on laboratory and literature data, provides an overview of the efficiency and impact of our system. | ||
<i>E. cowli</i> is capable of utilizing methane for the production of methanol. Methanol is subsequently excreted and metabolized by other organisms of the cows' microbiota [1]. To degrade methane <i>E. cowli</i> uses the well-characterized enzyme complex soluble methane monooxygenase (sMMO) from <i>M. capsulatus</i> catalyzing the conversion of methane to methanol with simultaneous consumption of oxygen and the cofactor NADH+H<sup>+</sup/> (see eq. 1 and eq. 2).<br></p> | <i>E. cowli</i> is capable of utilizing methane for the production of methanol. Methanol is subsequently excreted and metabolized by other organisms of the cows' microbiota [1]. To degrade methane <i>E. cowli</i> uses the well-characterized enzyme complex soluble methane monooxygenase (sMMO) from <i>M. capsulatus</i> catalyzing the conversion of methane to methanol with simultaneous consumption of oxygen and the cofactor NADH+H<sup>+</sup/> (see eq. 1 and eq. 2).<br></p> | ||
- | <img class="eq" src="http://latex.codecogs.com/gif.latex?\mathrm{CH_4}&space;+&space;\mathrm{O_2}&space;+&space;\mathrm{NADH}+\mathrm{H}^+&space;\rightarrow&space;\mathrm{CH_3OH}&space;+&space;\mathrm{H_2O}&space;+&space;\mathrm{NAD}^+&space;(1)" title="\mathrm{CH}_4 + \mathrm{O}_2 + \mathrm{NADH}+\mathrm{H}^+ \rightarrow \mathrm{CH_3OH} + \mathrm{H_2O} + \mathrm{NAD^+} | + | <img class="eq" src="http://latex.codecogs.com/gif.latex?\mathrm{CH_4}&space;+&space;\mathrm{O_2}&space;+&space;\mathrm{NADH}+\mathrm{H}^+&space;\rightarrow&space;\mathrm{CH_3OH}&space;+&space;\mathrm{H_2O}&space;+&space;\mathrm{NAD}^+&space;(1)" title="\mathrm{CH}_4 + \mathrm{O}_2 + \mathrm{NADH}+\mathrm{H}^+ \rightarrow \mathrm{CH_3OH} + \mathrm{H_2O} + \mathrm{NAD^+}&space; (1)" /><br> |
- | <img class="eq" src="http://latex.codecogs.com/gif.latex?\mathrm{MMO}&space;+&space;\mathrm{Me}&space;\overset{k_1}{\underset{k_-_1}{\rightleftharpoons}}&space;[\mathrm{MMO-Me}]\overset{k_2}{\rightarrow}&space;\mathrm{MeOH}&space;+&space;\mathrm{MMO}&space;(2)" title="\mathrm{MMO} + \mathrm{Me} \overset{k_1}{\underset{k_-_1}{\rightleftharpoons}} [\mathrm{MMO-Me}]\overset{k_2}{\rightarrow} \mathrm{MeOH} + \mathrm{MMO} | + | <img class="eq" src="http://latex.codecogs.com/gif.latex?\mathrm{MMO}&space;+&space;\mathrm{Me}&space;\overset{k_1}{\underset{k_-_1}{\rightleftharpoons}}&space;[\mathrm{MMO-Me}]\overset{k_2}{\rightarrow}&space;\mathrm{MeOH}&space;+&space;\mathrm{MMO}&space;(2)" title="\mathrm{MMO} + \mathrm{Me} \overset{k_1}{\underset{k_-_1}{\rightleftharpoons}} [\mathrm{MMO-Me}]\overset{k_2}{\rightarrow} \mathrm{MeOH} + \mathrm{MMO} &space;(2)" /><br> |
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Reported kinetic rates were determined at 18°C. However, the optimal temperature for the reaction has been reported to be 45°C which is within the temperature range for optimal growth of <i>M. capsulatus</i> [3]. Therefore, the rate kinetics were adjusted to the temperature during<i> in vitro </i>measurement and the actual temperature inside the cow’s rumen. These temperatures were 42°C for <i>M. capsulatus</i>, 37°C for <i>E. cowli</i> and 40°C for in vivo modelling. Based on the reaction kinetics proposed by Arrhenius (see eq. 3) the rate constants for various temperatures are determined. Estimated reaction rates are shown in table 2. | Reported kinetic rates were determined at 18°C. However, the optimal temperature for the reaction has been reported to be 45°C which is within the temperature range for optimal growth of <i>M. capsulatus</i> [3]. Therefore, the rate kinetics were adjusted to the temperature during<i> in vitro </i>measurement and the actual temperature inside the cow’s rumen. These temperatures were 42°C for <i>M. capsulatus</i>, 37°C for <i>E. cowli</i> and 40°C for in vivo modelling. Based on the reaction kinetics proposed by Arrhenius (see eq. 3) the rate constants for various temperatures are determined. Estimated reaction rates are shown in table 2. | ||
</p> <br> | </p> <br> | ||
- | <img class="eq" src="http://latex.codecogs.com/gif.latex?k&space;=&space;A&space;\cdot&space;e^{\frac{E_{A}}{R&space;\cdot&space;T}}&space;(3)" title="k = A \cdot e^{\frac{-E_{A}}{R \cdot T}} (3)"/><br><br> | + | <img class="eq" src="http://latex.codecogs.com/gif.latex?k&space;=&space;A&space;\cdot&space;e^{\frac{-E_{A}}{R&space;\cdot&space;T}}&space;(3)" title="k = A \cdot e^{\frac{-E_{A}}{R \cdot T}} (3)"/><br><br> |
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<div class="col-sm-5 col-md-6"> | <div class="col-sm-5 col-md-6"> | ||
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- | The calculated rate constants are comparable to values reported in the literature [5]. Based on the reaction rates the experimentally achieved data (see <a href="https://2014.igem.org/Team:Braunschweig/Results>our results page</a>) were fitted using the previously described Michaelis-Menten kinetics to estimate the initial concentration of the enzyme. The decrease of substrate concentration is described by eq. 4 and eq. 5.<br></p> | + | The calculated rate constants are comparable to values reported in the literature [5]. Based on the reaction rates the experimentally achieved data (see <a href="https://2014.igem.org/Team:Braunschweig/Results">our results page</a>) were fitted using the previously described Michaelis-Menten kinetics to estimate the initial concentration of the enzyme. The decrease of substrate concentration is described by eq. 4 and eq. 5.<br></p> |
<img src="http://latex.codecogs.com/gif.latex?\frac{dMe}{dt}&space;=&space;-&space;k_1 \cdot Me \cdot(E_0-\frac{E_0 \cdot Me}{Me+K_M})&space;+&space;k_1\cdot \frac{E_0 \cdot Me}{Me+K_M}&space;(4)" title="\frac{dMe}{dt} = - k_1 \cdot Me\cdot(E_0-\frac{E_0 \cdot Me}{Me+K_M}) + k_1 \cdot \frac{E_0 \cdot Me}{Me+K_M} (4)" class="eq"/><br> | <img src="http://latex.codecogs.com/gif.latex?\frac{dMe}{dt}&space;=&space;-&space;k_1 \cdot Me \cdot(E_0-\frac{E_0 \cdot Me}{Me+K_M})&space;+&space;k_1\cdot \frac{E_0 \cdot Me}{Me+K_M}&space;(4)" title="\frac{dMe}{dt} = - k_1 \cdot Me\cdot(E_0-\frac{E_0 \cdot Me}{Me+K_M}) + k_1 \cdot \frac{E_0 \cdot Me}{Me+K_M} (4)" class="eq"/><br> | ||
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- | Hence the initial enzyme concentration was determined as 5.3 µM and 1.7 µM for <i>E. cowli</i> and <i>M. capsulatus</i>, respectively. After taking cell count and molecular weight of sMMO into account, a total of 2500 enzymes per single bacterial cell is determined for <i>E. cowli </i>. The calculated initial enzyme concentrations are supported by literature and experimental data, wherein up to 100,000 enzymes per cells are reported for high and very high copy plasmids [6]. Therefore, the calculated enzyme concentration are considered as reasonable.<br><br> | + | Hence the initial enzyme concentration was determined as 5.3 µM and 1.7 µM for <i>E. cowli</i> and <i>M. capsulatus</i>, respectively. After taking cell count and molecular weight of sMMO into account, a total of 2500 enzymes per single bacterial cell is determined for <i>E. cowli</i>. The calculated initial enzyme concentrations are supported by literature and experimental data, wherein up to 100,000 enzymes per cells are reported for high and very high copy plasmids [6]. Therefore, the calculated enzyme concentration are considered as reasonable.<br><br> |
- | Due to safety concerns the methane concentration during <i>in vitro</i> measurements was kept below the flammability or explosivity level of 4.4 to 17 % (v/v) [4]. However, the natural atmosphere inside the rumen contains around 27% (v/v) | + | Due to safety concerns the methane concentration during <i>in vitro</i> measurements was kept below the flammability or explosivity level of 4.4 to 17 % (v/v) [4]. However, the natural atmosphere inside the rumen contains around 27% (v/v) methane and 0.8 % (v/v) oxygen [7], [8]. Therefore, our mathematical model is used for up-scaling and determination of methane degradation kinetics. Additional values such as the volume of the rumen and retention time were extracted from literature data. <br> |
Assuming that a cow’s rumen has an average size of 100 L, the molecular concentration of methane inside the rumen can be calculated based on the reported density of methane. Hence, the methane concentration inside the rumen is approximately 11.13 M. The total amount of enzyme needed to degrade 11.13 M of methane is 244 g ensuring a complete degradation of methane in one day. Considering that the retention time inside the rumen is on average 4 days, much less enzyme can be used for cost reduction [7].<br><br> | Assuming that a cow’s rumen has an average size of 100 L, the molecular concentration of methane inside the rumen can be calculated based on the reported density of methane. Hence, the methane concentration inside the rumen is approximately 11.13 M. The total amount of enzyme needed to degrade 11.13 M of methane is 244 g ensuring a complete degradation of methane in one day. Considering that the retention time inside the rumen is on average 4 days, much less enzyme can be used for cost reduction [7].<br><br> | ||
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Therefore we used a typical mass balance analysis to display the decrease. Assuming a steady-state system before the introduction of <i>E. cowli</i> the methane balance can be described using equation (6).<br><br></p> | Therefore we used a typical mass balance analysis to display the decrease. Assuming a steady-state system before the introduction of <i>E. cowli</i> the methane balance can be described using equation (6).<br><br></p> | ||
- | <img src="http://latex.codecogs.com/gif.latex?\frac{dMe_{Total}}{dt}&space;=&space;\frac{dMe_{Production}}{dt}&space;-&space;\frac{dMe_{Release}}{dt}&space;-&space;\frac{dMe_{Rumen}}{dt}&space;(6)" title="\frac{dMe_{Total}}{dt} = \frac{dMe_{Production}}{dt} - \frac{dMe_{Release}}{dt} - \frac{dMe_{Rumen}}{dt} (6)" class="eq" /><br> | + | <img src="http://latex.codecogs.com/gif.latex?\frac{dMe_{Total}}{dt}&space;=&space;\frac{dMe_{Production}}{dt}&space;-&space;\frac{dMe_{Release}}{dt}&space;-&space;\frac{dMe_{Rumen}}{dt}&space;(6)" title="\frac{dMe_{Total}}{dt} = \frac{dMe_{Production}}{dt} - \frac{dMe_{Release}}{dt} - \frac{dMe_{Rumen}}{dt} &space;(6)" class="eq" /><br> |
<img class="eq" src="http://latex.codecogs.com/gif.latex?\mathmr{with}~\frac{dMe_{Total}}{dt}&space;=&space;0" title="\mathmr{with} \frac{dMe_{Total}}{dt} = 0" /><br><br> | <img class="eq" src="http://latex.codecogs.com/gif.latex?\mathmr{with}~\frac{dMe_{Total}}{dt}&space;=&space;0" title="\mathmr{with} \frac{dMe_{Total}}{dt} = 0" /><br><br> | ||
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- | Degradation of methane took 3.5 times longer if <i>E. cowli</i> was immobilized (see | + | Degradation of methane took 3.5 times longer if <i>E. cowli</i> was immobilized (see figure 4). Based on Michaelis-Menten kinetics and our mathematical model an initial concentration of 2.3 µM was estimated corresponding to the amount of active enzyme. In comparison, the amount of initial, active enzyme in non-immobilized <i>E. cowli</i> has been 2.4 times higher. However, this was not unexpected. The beads were manually produced via polymerization of the alginate. Residual alginate as well as natural product loss led to the reduction of used cells and ultimately to a loss of enzyme. This was quantified by subsequently weighing of residual alginate. The loss amounted approximately 32 % due to an unoptimized method and a high viscosity of the alginate solution. Moreover, a possible explanation for the loss of activity lies in the variation of pH between the media. The activity of sMMO has been reported to be highly dependent on the milieu [11]. |
Fortunately, the pH of ruminal fluid fluctuates between 6.7 and 7.2 [7]. The measured pH values of the ruminal fluid and the NMS-media were 6.7 corresponding to a 36 % loss in activity. Thus, approximately only 43.52 % of the initial enzyme is active, representing the worst case scenario. An industrial production of alginate beads for entrapment is much more effective.<br><br> | Fortunately, the pH of ruminal fluid fluctuates between 6.7 and 7.2 [7]. The measured pH values of the ruminal fluid and the NMS-media were 6.7 corresponding to a 36 % loss in activity. Thus, approximately only 43.52 % of the initial enzyme is active, representing the worst case scenario. An industrial production of alginate beads for entrapment is much more effective.<br><br> | ||
- | Furthermore a discrepancy between cultivation of immobilized <i>E. cowli</i> in NMS-media and ruminal fluid was observed. The cultivation in NMS-media resulted in a slightly lower rate of methane degradation if compared to cultivation in rumen fluid, which is discussed in the <a href="https://2014.igem.org/Team:Braunschweig/Results-content#resultspart3"</a> | + | Furthermore a discrepancy between cultivation of immobilized <i>E. cowli</i> in NMS-media and ruminal fluid was observed. The cultivation in NMS-media resulted in a slightly lower rate of methane degradation if compared to cultivation in rumen fluid, which is discussed in the <a href="https://2014.igem.org/Team:Braunschweig/Results-content#resultspart3">results.</a><br> |
</p> | </p> | ||
<div class="fig-caption-right" | <div class="fig-caption-right" | ||
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<div class="fig-caption" style="width:100%;" align="center"> | <div class="fig-caption" style="width:100%;" align="center"> | ||
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- | <li><img style="border: 0px solid #dddddd; border-radius: 0px;" src="https://static.igem.org/mediawiki/2014/9/92/TU-BS_Kuh_gleich_Auto.png" alt="Cow = Car?"><a class="anchor" name="Cow = Car?" | + | <li><img style="border: 0px solid #dddddd; border-radius: 0px;" src="https://static.igem.org/mediawiki/2014/9/92/TU-BS_Kuh_gleich_Auto.png" alt="Cow = Car?"><a class="anchor" name="Cow = Car?"></li> |
<li>Does a cow equal a car?</li> | <li>Does a cow equal a car?</li> | ||
</ul> | </ul> |
Revision as of 22:38, 17 October 2014
Modeling Approach
Due to the increasing consumption of beef and dairy products cattle are nowadays a major contributor to the emission of greenhouse gases, thus vastly affecting global warming. In this year's project the iGEM Team Braunschweig is aiming at reducing the cows' share of the cake by designing a methane degrading bacterium – E. cowli.
However, due to safety and ethical concerns it is not easily manageable to test our system in vivo. Nonetheless, the effects of E. cowli on methane emissions by cattle need to be evaluated. Therefore, we created a mathematical model simulation based on data experimentally obtained in this project and previously published literature. The model was used to evaluate eventual costs and a theoretical scale-up of the system.
Mathematical Model
In this year's iGEM project, our objective is to decrease the amount of methane produced through enteric fermentation inside the cows’ rumen without affecting the internal microbiota. Produced methane is subsequently released from the digestive tract through the mouth by eructation or burping. To inhibit the emission, thus reducing the atmospheric methane levels, we established a methane degrading bacterium – E. cowli.
Our mathematical model, based on laboratory and literature data, provides an overview of the efficiency and impact of our system.
E. cowli is capable of utilizing methane for the production of methanol. Methanol is subsequently excreted and metabolized by other organisms of the cows' microbiota [1]. To degrade methane E. cowli uses the well-characterized enzyme complex soluble methane monooxygenase (sMMO) from M. capsulatus catalyzing the conversion of methane to methanol with simultaneous consumption of oxygen and the cofactor NADH+H+ (see eq. 1 and eq. 2).
According to literature data the reaction kinetics can be described using Michaelis-Menten kinetics [2]. Kinetic parameters varied from 3 to 23 µM for the Michaelis-Menten constant (KM), thus the most confident values shown in table 1 were selected for modeling.