Metabolism
This module simulates the processes of Respiration, Fermentation, and Photosynthesis. It also models the role of Photosynthetic Active Radiation (PAR) in photosynthesis and electron Donor-to-Acceptor ratio (eDAR) on respiration and fermentation. E. Coli bacteria carry out fermentation and respiration for energy production, while photosynthesis is carried out by an undefined primary producer. The chemical reactions for these three processes are:
Photosynthesis:
\[\begin{equation}\label{eq:Photosynthesis} 6 CO_2 + 6 H_2 O \rightarrow C_6H_{12}O_6 + 6 O_2 \end{equation}\]Cellular respiration:
\[\begin{equation}\label{eq:Respiration} C_6H_{12}O_6 + 6 O_2 \rightarrow 6 CO_2 + 6 H_2 O \end{equation}\]Ethanol fermentation:
\[\begin{equation}\label{eq:Fermentation} C_6H_{12}O_6 \rightarrow C_6H_{12}O_6 + 2 CO_2 \end{equation}\]These processes defined the state variables in this module: Glucose (\(C_6H_{12}O_6\)), Oxygen (\(6 O_2\)), Water (\(H_2O\)), Carbon Dioxide (\(6 CO_2\)), and Ethanol (\(C_6H_{12}O_6\)). Note that, for Oxygen and Carbon Dioxide the state variables correspond to the weight of the minimum number of molecules required for a single cycle of either metabolic pathway. The units of the state variables were g/ml. Although these units are uncommon in chemistry, they were consistent with biomass units of bacteria.
| State variable | molecular weight (g/mole) | weight of a molecule (g) |
|---|---|---|
| \(C_6H_{12}O_6\) | 180.156 | \(2.99\cdot 10^{-22}\) |
| \(O_2\) | 32 | \(5.31\cdot 10^{-23}\) |
| \(H_2O\) | 18.02 | \(2.99\cdot 10^{-23}\) |
| \(CO_2\) | 44.01 | \(7.31\cdot 10^{-23}\) |
| \(C_6H_{12}O_6\) | 46.07 | \(2.99\cdot 10^{-22}\) |
Rates of pathways and PAR
The next step was to define the rate at which Oxygen, Glucose, Carbon Dioxide, and Water were processed in the respective pathways. The \(O_2\) minimum and maximum consumption rates of E. Coli doing performing cellular respiration are Riedel et al, 2013 :
\(q^{min}_{O2}=1.67 \cdot 10^{-16}\) g/cell h
and
\(q^{max}_{O2}=1.33\cdot 10^{-13}\) g/cell h
By stoichiometry of cellular respiration we infer that minimum and maximum glucose consumption rates are \(q^{min}_{GR}=1.243 \cdot 10^{-16}\) g/cell h and \(q^{max}_{GR}=1.246 \cdot 10^{-13}\) g/cell h. That is, on a first approximation oxygen and glucose have to be consumed at the same rate because they are part of the same process, only in different quantities.
The fermentation rate was extracted from Seong, H.J et al, 2020:
\(q_{GF}=4.89 \cdot 10^{-13}\) g/cell h
See Calculations_for_parameters for a more detailed calculation.
The rate at which photosynthesis occurs is more complicated, because it partially depends on the Photosynthetic Active Radiation (PAR). PAR had its own submodule or container in the metabolic model:
The assumption is that the maximum photosynthetic rate of CO\(_2\) maximum photosynthetic rate is \(q^{max}_{CP}= 2.648 \cdot 10^{-6}\) g/ml h (data element ‘Max_photosynthetic_rate’ in the figure above). This rate was obtained from the estimation that a gram of leaf processes 44.14 ppm of CO\(_2\) per minute, converting ppm to mol/L and assuming a volume V=1 L. This is a very broad number, and needs to be refined in future iterations of the model.
On the other hand, primary productors need Photosynthetic Active Radiation (PAR) to carry out photosynthesis, so the actual rate at which CO\(_2\) will be \(q^{max}_{CP}\) only when PAR reaches its maximum. In fact, if PAR is below certain threshold, photosynthesis cannot be carried out. Here, we take the data in Ge et al. The data here corresponds to the San Francisco Bay Area. Also, the PAR values considered in this model ara averaged over a year. In other words, our model assumes that every day there is the exact same amount of PAR.
The raw data of PAR is given in units of mol/m\(^2\) h (moles of photons), so the first step is to convert moles of photons to Joules (J). This gives J/m\(^2\) h. We assume a photosynthetizing surface of m\(^2\) and get J/h (power) These calculations are done in the left hand side of the model in the figure above.
Considering \(q^{max}_{CP}\) and PAR the next step was to calculate the mass of CO\(_2\) metabolized from photosynthesis. This was done with the function element ‘Required_Consumption_CO2’. This function is the product of data element ‘Max_Photosynthetic_Rate’ (discussed above), data element ‘Photosynthetic_Biomass’ (weight of primary producers), and data element ‘Photosynthetic_Unit_Conversion’.
The next step was to impose upper and lower conditions. The upper condition comes from the limitation of PAR and is given by function element ‘Max_power_to_CO2’. This is the maximum CO\(_2\) that can be metabolized for a given amount of PAR. This element has the expression:
(Total_Power_Out/Activation_Energy_Phot)(6CO2_molecule_weight/1ml)
with ‘Activation_Energy_Phot’\(=121672.6\) meV. This expression has the following meaning: the power extracted from PAR divided by the Activation energy of photosynthesis gives the number of cycles of photosynthesis per hour. Multiplying this number by the weight of six molecules of CO\(_2\) gives the mass of CO\(_2\) metabolized per hour. (Note that this expression makes an assumption about the estimation of photosynthesis cycles). If ‘Required_Consumption_CO2’ is equal or larger than ‘Max_power_to_CO2’, only ‘Max_power_to_CO2’ can be metabolized. This condition is implemented by the selector function ‘Total_CO2_Consumption’.
The lower condition comes from the activation energy of photosynthesis (data element ‘Activation_Energy_phot’ in the figure above): if the power extracted from PAR is lower than the activation energy in an hour, photosynthesis will not ensue. This is implemented by the selector function ‘Activation_Photosynthesis’ in the figure above.
Respiration and Fermentation
Respiration and fermentation where implemented in the container ‘**Respiration_Fermentation’ as a set of function elements, as shown below:
As explained above, the consumption of O\(_2\) and C\(_6\)H\(_{12}\)O\(_6\) (glucose) was determined by the product of bacterial concentration (pool element ‘Bacteria’) and the consumption rate per bacterial cell or per capita (data elements ‘Per_capita_Bact_O2_intake’, ‘Resp_Per_cap_glucose’, and ‘Ferment_Per_cap_glucose’). This was encoded by function elements ‘Total_O2_Intake’, ‘Aerobic_Total_Glucose_Intake’, and ‘Anaerobic_Total_Glucose_Intake’. The total amount of glucose intake by both respiration and fermentation was encoded by the function element ‘Total_Glucose_Intake’.
The share of respiration and fermentation carried out by bacteria was a function of the electron-donor-to acceptor ratio (eDAR).
Mathematically, eDAR is:
\(\begin{equation} eDAR=\frac{[C_6 H_{12}O_6]}{[O_2]} \, , \end{equation}\) with the braces indicating concentrations.
This part of the model needs to be reviewed, because eDAR was implemented in GoldSim as:
\[\begin{equation} eDAR_{GS}=\frac{[O_2]}{[C_6 H_{12}O_6]} \, . \end{equation}\]This implies that \(eDAR=\frac{1}{eDAR_{GS}}\)
If eDAR\(_{GS}\) is very small, the concentration of O\(_2\) is much lower than the concentration of C\(_6\) H\(_{12}\)O\(_6\), indicating a very anaerobic environment. This is consisting with fermentation. If eDAR is high, the concentration of O\(_2\) is larger than that of C\(_6\) H\(_{12}\)O\(_6\). This is consistent with an aerobic environment where respiration (generally) dominates.
To formalize the connection between environmental conditions and eDAR, we use a Hill function see Model 6 for a more detailed discussion. In the GoldSim representation, function element ‘HeDAR’ encodes the Hill function. HeDAR takes values between 0 (only fermentation) and 1 (only respiration). The hill exponent was set to \(n=7\) to have a more abrupt change at intermediate values of eDAR. Future versions of this model should implement the modified Hill function discussed in Model 6.
The Hill function was linked to fermentation and respiration by the function elements ‘Total_Glucose_Intake’ and ‘Total_O2_Intake’.
![Metabolic_Machine]Metabolic_Machine_edar_to_resp_ferm.PNG “Courtesy of GoldSim”)
‘Total_Glucose_Intake’ has the formula:
HeDARAnaerobic_Total_Glucose_Intake + (1-HeDAR)Aerobic_Total_Glucose_Intake
and ‘Total_O2_Intake’:
(1-HeDAR)Per_capita_Bact_O2_intakeBacteria.
The delay element ‘MaterialDelay_Gluc’ is very important. It ensures that GoldSim is not outflowing more glucose than there actually is in the ‘Glucose’ pool element.