Objectives

The objectives of our mathematical model are to:

• Demonstrate and prove the feasibility of our project design by showing how adding human carbonic anhydrase II (abbreviated as CA) affects the consumption of carbon dioxide by the cyanobacteria, synechococcus elongatus, as opposed to natural conditions without CA.
• Predict how the indoor CO2 concentration changes over time, given the number of people in the room, the exact volume of the room, and whether or not CA is utilized.
• Determine the theoretical maximum number of people allowed in a room to maintain the indoor CO2 concentration under a certain level, with our product utilized.

Our model is based on two major assumptions. For the sake of simplicity, we assume that the space under examination is enclosed and airtight, meaning that inflow to and outflow from the room are both eliminated. In addition, regarding the air conditioning, which is a major source of air flow, since internal air is recirculated without interacting with external air with most modern air conditioning systems, it can be assumed that air conditioning does not affect the indoor carbon dioxide level.

OVERVIEW OF MODEL PARAMETERS

Several factors influence the CO2 concentration in an enclosed space (for example: a meeting room), namely the following:

1. Additive:
   • The number of people in the space.
   • The average CO2 exhalation rate per person.
   • The average CO2 concentration in exhale.
2. Subtractive:
   • The natural dissolution rate of CO2 in water (when CA is absent).
   • The dissolution rate of CO2 in water when CA is present, expressed as the enzymatic reaction rate of CA.
   • The rate of CO2 consumption from the photosynthesis of cyanobacteria.
   • The solubility of CO2 in water at various acidities.
3. Other:
   • The volume of the enclosed space.
   • Initial indoor CO2 concentration.

We obtained literature values and models for Factors 1b, 1c, and 2. Factors 1a and 3 are treated as independent variables in the model that can be input accordingly to calculate the indoor CO2 concentration.

Factor 1: Modeling the CO2 produced by human exhalation

In order to come up with a comprehensive mathematical model, we decided to start by modeling the indoor CO2 concentration over time, considering human breathing. The indoor CO2 concentration in this case can be modeled with the following formula (Jung).

Equation 1. The governing equation modeling the concentration of carbon dioxide (C) in an enclosed space at a specific time (t), given the number of people in the room (n), the CO2 concentration in exhale (C_ex), the exhalation flow rate (Q_ex), the volume of the room (V_room), and the initial indoor CO2 concentration (C₀).

In order to better combine this equation with other factors of our model, we rewrote Equation 1 into its differential form and defined it as the instantaneous rate of human production of CO2 by exhalation at time t, v_human(t).

Equation 2. The instantaneous change in indoor CO2 concentration (derivative of Equation 1).

Factor 1b: Determining the Exhalation Rate of an Individual (Q_ex)

Exhalation rate is the amount of air that an individual exhales per unit time. From literature, the exhalation rate of an individual is shown below (Jung). The default value used in our model is calculated by the average of minimum and maximum values. Unit conversions are shown below.

Table 1. Exhalation rate of an individual with unit conversion(Jung)

Factor 1c: Determining the CO2 Concentration in Exhale (C_ex)

The CO2 concentration of an individual's breathing varies highly and is dependent on the individual and their condition. The CO2 concentration in exhale from literature is shown below (Jung). The average of minimum and maximum values is used as the default value in our model. Unit conversions are shown below, with assumptions of an environment with an air pressure of 100 kPa and an air temperature of 25 ℃ (or 297 K).

Table 2. CO2 concentration in exhale with unit conversion (Jung)

Factor 2: Modeling the Factors that Reduce Carbon Dioxide

The CO2 fixation process can be illustrated with the following diagram.

Therefore, the process of defining the sequestration of CO2 in our model can be divided into two aspects: determining the natural dissolution rate of CO2 in water (when CA is absent) and determining the increased dissolution rate of CO2 in water (when CA is present). However, the dissolved inorganic carbon (DIC; which implies the combination of H₂CO₃, HCO₃^-, _CO3^2-, and all other forms of dissolved carbon dioxide) in water can eventually reach the maximum capacity, which is itself dependent on the pH value of the environment. How rapidly this theoretical maximum is attained is dictated by the photosynthetic rate of cyanobacteria.

Factor 2a: Determining the Natural Dissolution Rate of CO2 in Water

The natural dissolution rate of CO2 in water (v_natural) is calculated using the first-order rate constant equation, as shown in Equation 3, where 0.039 s^-1 is the rate constant for the forward reaction of the hydration of carbon dioxide (CO₂+H₂O->H₂CO₃). The correlation between the concentration of gaseous carbon dioxide and the natural dissolution rate is illustrated in Figure 1.

Equation 3. First-order rate constant equation for hydration of carbon dioxide.

Figure 1. The linear curve modeling the correlation between the concentration of gaseous carbon dioxide and its natural dissolution rate in water.

Factor 2b: Determining the Enzymatic Reaction Rate of CA

We modeled the enzymatic rate of carbonic anhydrase with Michaelis-Menten kinetics, as shown in Equation 4. v_CA represents the enzymatic reaction rate of CA; [CO_2(g)] represents the concentration of substrate (in this case, gaseous carbon dioxide); v_max represents the maximum reaction rate possible; and K_m is the Michaelis-Menten constant, which is defined as the substrate concentration when the enzymatic reaction rate is half of its maximum (whenv=1/2v_max).

Equation 4. The Michaelis-Menten equation, which models the relationship between the concentration of gaseous carbon dioxide and the enzymatic reaction rate.

Equation 5. Calculation for v_max, the maximum reaction rate. K_cat denotes the catalyst rate constant of CA and [CA] represents the concentration of the enzyme.

Therefore, by inserting the v_max and K_m values into the model, the correlation between CO2 concentration and the reaction rate of CA can be shown. v_max can be calculated with Equation 5.
We obtained the K_cat and K_m values for the human carbonic anhydrase II (hCAII) from literature. According to Silverman et al., the K_cat and K_m values for hCAII are 1.4×10⁶ s^-1 and 9.3×10^-3 M, respectively. A [CA] value of 2.883×10^-3 M is used based on the enzyme we use. Unit conversions are shown below.

Using the enzymatic coefficients for the Michaelis-Menten model, we plotted the curve in Figure 2 to model the enzyme kinetics of human carbonic anhydrase II.

Figure 2. The Michaelis-Menten curve modeling the enzyme activities of hCAII at various concentrations of carbon dioxide.

Factor 2c: Determining the Photosynthetic Rate of Cyanobacteria in the Form of CO2 Consumption

After carbon dioxide has fully hydrated and dissolved into water, the last phase in our process of CO2 sequestration is its removal by the photosynthesis of cyanobacteria Synechococcus elongatus PCC7942. The rate at which the dissolved CO2 is consumed by our cyanobacteria can be modeled with Equation 5, which models the relationship between the concentration of dissolved inorganic carbon (DIC) and the rate of carbon-specific growth (Cμ; which is also equivalent to the rate of photosynthesis v_{photosynthesis}), which is represented in the form of O2 production (Clark). The conversion from the amount of O2 production to CO2 sequestration is then performed using the 1:1 molar ratio between O2 and CO2 obtained from the chemical equation of photosynthesis. The maximum rate of growth μ_max and the half saturation constant for C_μ, K_G(DIC), specific for the cyanobacteria Synechococcus are then consulted from literature. Calculations for the final function are shown below with the parameters listed in Table 3.

Equation 6. Relationship between the rate of photosynthesis (v_{photosynthesis}) and the concentration of dissolved inorganic carbon (DIC) (Clark).

Table 3. Parameters used in calculation for C_μ

Using the converted photosynthetic coefficients that we obtained through literature for our cyanobacteria Synechococcus elongatus, we constructed a graph in Figure 3 showing the relationship between the concentration of dissolved inorganic carbon in water and the rate of CO2 consumption due to photosynthesis.

Figure 3. The Michaelis-Menten curve modeling the photosynthetic activity of Synechococcus elongatus. The CO2 consumption rate at various concentrations of dissolved inorganic carbon (DIC) is shown.

Factor 2d: Determining the Solubility of CO2 in Alkaline Environment

In order to model the maximum dissolvable amount of carbon dioxide in alkaline environments, we first used the Henry's Law (Equation 7), which states that the concentration (C) of a gas in liquid is equal to its partial pressure (P_gas) multiplied by the Henry's law solubility constant (H^cp).

Equation 7. Henry's Law, modeling the relationship between the partial pressure of gas and its concentration in a solvent.

From Sander (2015), the Henry's law constant (H^cp) of carbon dioxide at 25 °C with water as a solvent is 3.3 ⨉ 10^-4 mol·m^-3·Pa^-1. Unit conversion is shown below.

Assuming the concentration of CO2 in air is 400 ppm and the air pressure is 1 atm, the concentration of CO2 in water can be calculated as the following:

With the rates of the solubility of CO2 identified both with or without the addition of CA, it is crucial to note that an enzyme does not increase the final yield of products, but instead accelerates its reaction rate. Therefore, it is critical to identify the maximum attainable concentration in liquid. It is also essential to note that the solvent we use in our project is a culture medium for Synechococcus elongatus and is alkaline with a pH value of 11. The effect of pH on solubility is summarized by the following equation, with the maximum concentration from Henry’s Law considered as [CO_{2 (aq)}]:

Our Final Model

References