Modelling Approach

Due to the increasing consume 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 affection 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. To degrade methane E. cowli uses the in M. capsulatus well-characterized enzyme complex soluble methane monooxygenase (sMMO) 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 (K_M), thus the most confident values shown in table 1 were selected for modeling.

Parameter	Value	Reference
KM (CH4)	23 µM	[2]
k1	2 &times 108 M/s	[3]
k-1	1 &times 102 s-1	[3]
kcat	6 s-1	[3]
Activation Energy	formation of intermediate 13.8 kcal	[4]
Activation Energy	decay of intermediate 8.4 kcal	[4]

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 M. capsulatus [3]. Therefore the rate kinetics were adjusted to the temperature during in vitro measurement and the actual temperature inside the cow’s rumen. These temperatures were 42°C for M. capsulatus, 37°C for E. cowli and 40°C for in vivo modelling. Based on the reaction kinetics proposed by Arrhenius (see eq. 3) the rate constant for various temperatures are determined. Estimated reaction rates are shown in table 2.

Temperature [°C]	k1[ M/s]	k2 [s-1]
18 [2]	2 &times 108	6
37	8.6791 &times 108	14.5485
40	1.0766 &times 109	16.5686
42	1.2401 &times 109	18.0441

The calculated rate constants are comparable to values reported in literature [5]. Based on the reaction rates the experimentally achieved data (see results) 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.

Hence the initial enzyme concentration was determined as 5.3 µM and 1.7 µM for E. cowli and M. capsulatus, respectively. The calculated initial enzyme concentrations are supported by literature and experimental data, wherein an average of 2500 enzymes per single bacterial cell is reported for high copy plasmids [6]. Due to safety concerns the methane concentration during in-vitro 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 upscaling and determination of methane degradation kinetics. Additional values such as the volume of the rumen and retention time were extracted from literature data. Assuming that a cow’s rumen has an average size of 100 L, the molecular concentration of methane inside the rumen can be calculated as 11.13 mol/L. The total amount of enzyme needed to minimize the emission 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].

However, up to here the mathematical model only includes the decrease of methane inside the rumen. For evaluation of the environmental impact as well as the effects on methane emission it is important to know how much less methane will be released into the ambient air. Therefore we used a typical mass balance analysis to display the decrease. Assuming a steady-state system before the introduction of E. cowli the methane balance can be described using equation 6.

with dMe_{Total}=0

Considering that gases are infinitely soluble in other gases, we can assume that the reported release of approximately 300 g methane per day also corresponds to 27 % (v/v) methane in the eructation [9]. Due to the balance between ruminal and released methane based on the solubility of gases in each other, the degradation of methane immediately affects its release into the environment. Thus from the very first second less methane will be emitted.

The final application of our project is ideally the introduction of E. cowli into the cows rumen referring to a degradation of methane at its source. Contrary to previous approaches without the internal microbiota of the cow is not affected. (LINK!!) However an important issue of this approach is the viability of E. cowli inside the rumen (LINK!!). To overcome this issue an immobilisation of E. cowli inside calcium alginate beads was performed (Link..). The calcium alginate matrix mainly consists of water, therefore a diffusion limitation was neither visible in the experimentally obtained data nor in literature [10] LINK ZUU DEN LILA BEADS. However a decrease in methane degradation after immobilization was observed compared to free bacteria in suspension.

Degradation of methane took 3.5 times longer if E. cowli was immobilized (see FIG ). 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 E. cowli 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 reduction of used cells and ultimately to a loss of enzyme. This was quantified by subsequent 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 szenario. Furthermore a discrepancy between cultivation of immobilized E. cowli 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 NMS-media, which is discussed in the results (LINK ZU DEN RESULTS)

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