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).

$\mathrm{CH}_4 + \mathrm{O}_2 + \mathrm{NADH}+\mathrm{H}^+ \rightarrow \mathrm{CH_3OH} + \mathrm{H_2O} + \mathrm{NAD^+}&space; (1)$
$\mathrm{MMO} + \mathrm{Me} \overset{k_1}{\underset{k_-_1}{\rightleftharpoons}} [\mathrm{MMO-Me}]\overset{k_2}{\rightarrow} \mathrm{MeOH} + \mathrm{MMO} &space;(2)$

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.

Table 1: Kinetic parameters for MMO from literature.

Parameter	Value	Reference
K_M (CH₄)	23 µM	[2]
k₁	2 × 10⁸ M^-1s^-1	[3]
k_-1	1 × 10² s^-1	[3]
k_cat	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 constants for various temperatures are determined. Estimated reaction rates are shown in table 2.

$k = A \cdot e^{\frac{-E_{A}}{R \cdot T}} (3)$

Figure 1: Calculated kinetic parameters obtained using the Arrhenius equation.

Table 2: Theoretically obtained kinetic parameter.
Temperature [°C] k₁[M^-1s^-1] k₂ [s^-1]

18 [2] 2 × 10⁸ 6

37 8.6791 × 10⁸ 14.5485

40 1.0766 × 10⁹ 16.5686

42 1.2401 × 10⁹ 18.0441

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

$\frac{dMe}{dt} = - k_1 \cdot Me\cdot(E_0-\frac{E_0 \cdot Me}{Me+K_M}) + k_1 \cdot \frac{E_0 \cdot Me}{Me+K_M} (4)$
$\frac{dMeOH}{dt}= k_2 \cdot \frac{E_0 \cdot Me}{Me+K_M} (5)$

Figure 2: Comparison of obtained experimental data for M. capsulatus and E. cowli. Experimentally achieved data is fitted using our mathematical model and Michaelis-Menten kinetics.

Hence the initial enzyme concentration was determined as 5.3 µM and 1.7 µM for E. cowli and M. capsulatus, respectively. After taking cell count and molecular weight of sMMO into account, a total of 2500 enzymes per single bacterial cell is determined for E. cowli. 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.

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 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.
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].

However, up to here the mathematical model only includes the decrease of methane inside the rumen. For the 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).

$\frac{dMe_{Total}}{dt} = \frac{dMe_{Production}}{dt} - \frac{dMe_{Release}}{dt} - \frac{dMe_{Rumen}}{dt} &space;(6)$
$\mathmr{with} \frac{dMe_{Total}}{dt} = 0$

Figure 3: Possible reduction of internal methane concentration and release of methane after application of E. cowli

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 the internal microbiota of the cow is not affected. However an important issue of this approach is the viability of E. cowli inside the rumen. To overcome this issue an immobilisation of E. cowli inside calcium alginate beads was performed. The calcium alginate matrix mainly consists of water, therefore a diffusion limitation was neither visible in the experimentally obtained data nor in literature [10]. However a decrease in methane degradation after immobilization was observed compared to free bacteria in suspension.

Figure 4: Decay in methane concentration of free and immobilized E. cowli in varying media. Purple: Immobilized in NMS-media, green: Immobilized in ruminal fluid and red: free bacteria in NMS-media.

Degradation of methane took 3.5 times longer if E. cowli 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 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 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.

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 rumen fluid, which is discussed in the results.

Figure 5: pH-dependent activity of sMMO (modified figure) [7].

The immobilization of E. cowli in a calcium alginate matrix allows growth and enzymatic activity while cultivated in ruminal fluid. The costs for production of alginate beads shall be evaluated subsequently. As previously determined a total active enzyme amount of 244 g is necessary for complete degradation of methane produced through enteric fermentation. In this project beads were manually produced using transfer pipettes, thus the diameter was 7 mm in average with 4×10⁹ cells. Consequently, a total of 750 beads is needed to reduce the methane emission to a minimum, which have a retention time of 4 days in the cows rumen. For the reduction of the costs, the bead composition can be optimized. It has been reported that small amounts of paraffin increase the solubility of methane in the liquid phase vastly [12]. However, the health effects of paraffin has to be evaluated.

Our costs for the production were 50 cent per ratio, which consists of 750 beads. Hence, a total of 50 $ per year is needed to reduce the annual methane emission by 110 kg per cow. Ideally the worldwide methane emission is reduced by 164 million tons, based on a total number of 1.5 billion cows [13]. This is the ideal case. Nevertheless in case only 1% percent of the sMMO is active if immobilized and introduced into the rumen, the methane emission is reduced by 164 ×10⁴ tons.

Considering the 25 times greater impact on global warming of methane in comparison to carbon dioxide, a comparative statistical analysis of cows with average emission values of cars is possible.

Does a cow equal a car?

Assuming that a cow releases a maximum of 500 L of methane and a minimum of 300 L per day, the annual emissions of carbon dioxide equivalents range from 2.737 to 4.562 t. An average car, consuming a maximum of 150 g and a minimum of 90 g of carbon dioxide per kilometer, covers in average a distance of 15.000 km per year. Hence, the annual emission rates range from 1,350 to 2,250 tons CO₂. Consequently, the cow’s impact on global warming is twice as great as the impact of a car.

Industrial application of this years iGEM project is able to reduce the annual methane emission by 110 kg methane per cow corresponding to 2750 kg CO₂-equivalent emissions.