Production-Release Model
Overview
To demonstrate the effectiveness of BOROHMA, we combined two models to simulate the production and release efficiency of minicell-encapsulated L-borneol.
In the first model, we used enzyme kinetics to understand the relationship between enzyme activity and substrate concentration. This model provides a framework to quantify how changes in substrate concentration affect the rate of reactions. By modeling the pathways involved in L-borneol production, we can simulate the production process and find our final yield based on the initial substrate concentration. Understanding this relationship will allow us to modify our production process and fine-tune L-borneol yield for maximum repellency and better aromas.
For our second model, since we're using minicell encapsulation to release L-borneol, it's essential to consider the transmembrane diffusion rate to assess how encapsulation influences both L-borneol release and its effective repellency. Since transmembrane diffusion rate is affected by the internal concentration of L-borneol, our second model utilizes the final yield obtained from our first model. This thorough understanding of L-borneol’s transmembrane movement is key to optimizing its production for maximum effectiveness.
Production: Kinetics
Method
There are two main pathways—the MVA pathway and the MEP pathway—and three downstream steps in the L-borneol production process. The following figure shows the entire reaction pathway for the production of L-borneol.
Figure 1. Overall pathway for L-borneol production
Glucose is the starting substrate in our pathway. However, as the pathway progresses, glucose is converted into various intermediates through a series of enzyme-substrate reactions. Each step is catalyzed by a specific enzyme that transforms the substrate into a new product, which then serves as the substrate for the next reaction. Eventually, L-borneol is produced.
Since the rate-limiting steps occur at a much slower rate than the non-rate-limiting steps, we assumed that the rate of the non-rate-limiting steps are negligible. Thus, we only modeled the rate-limiting steps, which are indicated by the red (overexpressed) enzymes shown in Figure 1. By chaining together the rate-limiting steps in the L-borneol production pathway, we are able to accurately approximate the rate of production of L-borneol. Our simplified version of the pathway only contains the rate-limiting steps, as seen in Figure 2.
Figure 2. The simplified version of our production pathway with only the rate-limiting steps (Clendening et al., 2010; Movahedi et al., 2021)
We utilized two different types of Michaelis-Menten equations in our production model. The first shows the equation for systems that contain only one substrate while the second describes a bi-substrate system.
\[ v = \frac{V_{\text{max}} \cdot [S]}{[S] + K_m} \]
\[ v = \frac{V_{\text{max}} [S_1] [S_2]}{K_m^{S_1} K_m^{S_2} + K_m^{S_1} [S_2] + K_m^{S_2} [S_1] + [S_1] [S_2]} \]
Variables
\( v \): production rate at a given time (unit: \( \mu M/sec \))
\( K_m \): the Michaelis-Menten rate constant. It is the substrate concentration when the
reaction rate is half the maximum velocity. (unit: \( \mu M \))
\( V_{\max} \): the fastest rate the reaction can reach before the substrate concentration no longer affects the velocity of the reaction (unit: \( \mu M/sec \)).
Assumptions
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Only the rate-limiting steps in the MEP pathway affect the overall rate of production.
- The MEP pathway occurs at a slower rate than the MVA pathway.
- Since the end products of both the MEP and MVA pathways are IPP and DMAPP and the enzyme IDI regulates their equilibrium, one pathway can be prioritized.
- More enzymes are overexpressed in the MEP pathway, further indicating that it may occur at a slower rate.
-
IPP and DMAPP are treated as a single substrate
- The IDI enzyme regulates the equilibrium between IPP and DMAPP and maintains a constant concentration ratio between the two substrates.
-
Absence of inhibition
- We assume that no inhibitors are present to prevent the binding of substrates to enzymes, allowing for more straightforward enzyme-substrate interactions.
-
G3P does not contribute to the production of Pyruvate
- G3P only contributes to the formation of DXP.
- G3P and Pyruvate are independent, non-interactive substrates.
- As both G3P and Pyruvate act as substrates in the first rate-limiting reaction of the MEP pathway, conversion of G3P to Pyruvate would overcomplicate enzyme kinetics equation for the reaction G3P + Pyruvate → DXP.
Michaelis-Menten Reaction Chain
We used the following equation to model the changes in substrate concentration over time for each step of the production of L-borneol.
\[ \frac{d[\text{Product}]}{dt} = \text{rate of formation} - \text{rate of consumption} \]
By implementing the equation above to each rate-limiting step in our pathway, we derive the chain of Michaelis-Menten equations below, with L-borneol as the final product. The only exception is the rate of formation of DXP, which follows a bi-substrate Michaelis-Menten equation, as both G3P and Pyruvate serve as substrates in the reaction (Dalwadi et al., 2018).
\[ \frac{d[DXP]}{dt} = \frac{V_{1max} [G3P][Pyruvate]}{K_{1m} K_{2m} + K_{1m} [Pyruvate] + K_{2m} [G3P] + [G3P] [Pyruvate]} - \frac{V_{2max} [DXP]}{[DXP] + K_{3m}} \]
\[ \frac{d[IPP \& DMAPP]}{dt} = \frac{V_{2max} [DXP]}{[DXP] + K_{3m}} - \frac{V_{3max} [IPP \& DMAPP]}{[IPP \& DMAPP] + K_{4m}} \]
\[ \frac{d[GPP]}{dt} = \frac{V_{3max} [IPP \& DMAPP]}{[IPP \& DMAPP] + K_{4m}} - \frac{V_{4max} [GPP]}{[GPP] + K_{5m}} \]
\[ \frac{d[BPP]}{dt} = \frac{V_{4max} [GPP]}{[GPP] + K_{5m}} - \frac{V_{5max} [BPP]}{[BPP] + K_{6m}} \]
\[ \frac{d[L\text{-borneol}]}{dt} = \frac{V_{5max} [BPP]}{[BPP] + K_{6m}} \]
Obtaining Constants
Obtaining the constants in our Michaelis-Menten equations, \( K_m \) and \( V_{\max} \), for each rate-limiting step proved to be a significant bottleneck to this model. While we were able to find most of the \( k_{\text{cat}} \) and \( K_m \) values for the MVA and MEP pathways from literature or enzyme databases such as BRENDA, there existed discrepancies between available data and our desired experimental conditions. These conditions include the environment in which the experiment was conducted (such as temperature and pH), the organism used, and the specific substrate involved in the reaction. In the calculations that we have done, we only utilized the constants that we believe are accurate for our model.
In addition to the kinetic constants, the value of \( V_{\max} \) is essential for our equations, and it can be calculated with the equation:
\[ V_{\max} = k_{\text{cat}} \cdot [E_t] \]
Since we obtained the values of \( k_{\text{cat}} \) through research, the only missing variables are the total concentrations of the rate-limiting enzymes.
We determined the total enzyme concentration by running a 10% SDS page for all the enzymes utilized in the system. Using a computer software called ImageJ, we measured the relative luminescence of each rate-limiting enzyme in comparison to the total luminescence for the lane. As a reference, we added a known quantity of Bovine Serum Albumin (BSA), 1 µg in 5 µL, to each enzyme sample. Below is an image of the SDS page.
Figure 3. Annotations of the enzymes on the 10% SDS page
We found that BSA has a concentration of 3.01 µM. To find the concentration for the other enzymes, we normalized the ratio for BSA and the other enzymes as we already know the concentration for BSA. This process was repeated for two induced trials, which were then averaged to get the final concentration to ensure higher accuracy. This is the calculation performed to obtain the concentration of BSA:
\[ 1 \times 10^{-6} \, \text{g} \times \frac{1 \, \text{mol}}{66500 \, \text{g}} \times \frac{1}{5 \times 10^{-6} \, \text{L}} = 3.01 \times 10^{-6} \, \text{M} = 3.01 \, \mu \text{M of BSA} \]
We obtained the relative ratio for each enzyme in comparison to BSA by getting the luminescence of the enzyme and dividing it by the luminescence of BSA. We then used the ratio and multiplied it by the known concentration for BSA. The following is the calculation for HmgR:
\[ \frac{1202287.166}{1049993.663} \times 3.01 \, \mu M = 3.45 \, \mu M [\text{DXS}] \]
The following list shows the total enzyme concentration for each rate-limiting enzyme:
Table 1. The corresponding \( K_m \), \( k_{\text{cat}} \), \( V_{\max} \), and concentration values for each rate-limiting reaction (Koppisch et al., 2002)
In addition, we can use the \( K_m \) and \( V_{\max} \) values to evaluate the enzyme's efficiency in catalyzing the reaction. Catalytic efficiency is calculated through \(\frac{k_{\text{cat}}}{K_m}\).
- When \( K_m \) is large, the enzyme has a low affinity and therefore can only reach \( V_{\max} \) at a high substrate concentration, making the reaction less efficient.
- When \( K_m \) is low, the enzyme has a high affinity and therefore can reach \( V_{\max} \) at a lower substrate concentration, making the reaction more efficient.
- When \( k_{\text{cat}} \) is large, the catalytic efficiency is high because the rate for product formation, given by \( \frac{d[P]}{dt} = k_{\text{cat}} [\text{ES}] \), is higher. This indicates that the enzyme is more efficient in catalyzing the reaction to form the product.
- When \( k_{\text{cat}} \) is small, the catalytic efficiency is low because the rate for product formation, given by \( \frac{d[P]}{dt} = k_{\text{cat}} [\text{ES}] \), is lower. This indicates that the enzyme is less efficient in catalyzing the reaction to form the product.
To summarize, if \( K_m \) is high or \( k_{\text{cat}} \) is low, the reaction is less efficient while if \( K_m \) is low or \( k_{\text{cat}} \) is high, the reaction is more efficient. This supports our assumption that the rate-limiting steps in the MEP pathway more significantly influence the overall production rate of L-borneol. The higher \( K_m \) value for the rate-limiting step in the MEP pathway (90 to 140 µM) compared to the MVA pathway (20 µM) further indicates that the MEP pathway is less efficient.
Model Simulation
After obtaining our constants, we solved our set of differential equations using Python, which allowed us to calculate the overall production rate of L-borneol and generate a graph showing the concentration changes of substrates over time at different steps of the pathway. However, it is important to note that several constants inputted in our code are placeholders. Despite not having specific kinetic parameters, the graph generated by our model simulation helps us to understand the relationship between the initial glucose concentration and the final yield of L-borneol. This allows us to determine the optimal initial glucose concentration needed to achieve our desired amount of L-borneol.
Our Python code uses the ODEINT function to solve the system of ordinary differential equations (ODEs). The reaction chain models how each variable interacts with one another in the overall system. The solver returns the substrate concentrations at specific time points, then, integrates these values into the next equation at subsequent time points (100 time points within 10-second intervals), capturing how each substrate concentration changes over time.
def michaelis_menten(S, Vmax, Km):
return Vmax * S / (Km + S)
def michaelis_menten_2vars (Sa, Sb, V2max, Kam, Kbm):
return V2max*Sa*Sb/(Kam*Kbm +Kam*Sa + Kbm*Sa + Sa*Sb)
def reaction_chain(y, t, V1max, K1m, K2m, V2max, K3m, V3max, K4m, V4max, K5m, V5max, K6m):
G3P, Pyruvate, DXP, IPPandDMAPP, GPP, BPP, Lborneol = y
# Rate of G3P +Pyruvate -> DXP
rateA = michaelis_menten_2vars(G3P, Pyruvate, V1max, K1m, K2m)
# Rate of DXP -> IPP and DMAPP
rate1 = michaelis_menten(DXP, V2max, K3m)
# Rate of IPP and DMAPP -> GPP
rate2 = michaelis_menten(IPPandDMAPP, V3max, K4m)
# Rate of GPP -> BPP
rate3 = michaelis_menten(GPP, V4max, K5m)
# Rate of BPP -> L-borneol
rate4 = michaelis_menten(BPP, V5max, K6m)
# ODEs
dG3P_dt = - 1*rateA
dPyruvate_dt = - 1*rateA
dDXP_dt = rateA - rate1
dIPPandDMAPP_dt = rate1 - rate2
dGPP_dt = rate2 - rate3
dBPP_dt = rate3 - rate4
dLborneol_dt = rate4
# Time points (0 to 10 seconds, 100 steps)
t = np.linspace(0, 10, 100)
# Solve ODEs, input with constants e.g. Vmax & Km
solution = odeint(reaction_chain, initial_concentrations, t, args=(V1max, K1m, K2m, V2max, K3m, V3max, K4m, V4max, K5m, V5max, K6m))
Figure 4. This graph depicts the concentration of rate-limiting substrates over time(constants found using XJTU-China's ProtGPT-2 model)
Limitations
There are multiple limitations that we were unable to model:
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Neglecting inhibition when the enzyme-substrate complex is bounded by an inhibitor
- Inhibition can affect the rate of production by temporarily or permanently reducing enzyme activity and halting the reaction. To obtain more accurate results, an extended Michaelis-Menten model that incorporates the effects of inhibitors is needed.
-
Inability to determine multiple kinetic parameters
- Insufficient research was done on our engineered downstream steps, so there are limited available literature data on the values of constants.
- We utilized the BRENDA database; however, each recorded constant is obtained under specific conditions that do not perfectly align with our experiments.
- Due to insufficient technology, we were unable to determine the kinetic constants from experimental data.
- Further exploration of constants through experimental data would allow us to more accurately model the rate of production of L-borneol.
Future Experiments
Since we were unable to find the necessary constants for modeling the production of L-borneol from literature, our future plan is to obtain these constants using experimental data. There are two regression models we could use to find our constants: the Lineweaver Burk plot and the Eadie Hofstee plot. Both plots require experimental data of different substrate concentrations and their corresponding initial reaction rates (Blaber, 2021).
Lineweaver Burk Plot
\[ \begin{aligned} & \frac{1}{v}=\frac{K_m+[S]}{V_{\max }[S]} \\ & \frac{1}{v}=\left(\frac{K_m}{V_{\max }}\right)\left(\frac{1}{[S]}\right)+\frac{1}{V_{\max }} \\ & y=m x+b \end{aligned} \]
\[ \begin{aligned} & y = \frac{1}{v} \\ & m = \frac{K_m}{V_{\max }} \\ & x = \frac{1}{[S]} \\ & b = \frac{1}{V_{\max }} \\ & \text{y-intercept: } \frac{1}{V_{\max }} \\ & \text{x-intercept: } -\frac{1}{K_m} \\ & \text{slope: } \frac{K_m}{V_{\max }} \end{aligned} \]
Eadie-Hofstee Plot
\[ \begin{aligned} & y = m x + b \\ & v = -K_m\left(\frac{v}{[S]}\right) + V_{\max} \end{aligned} \]
\[ \begin{aligned} & y = v \\ & m = -K_m \\ & x = \frac{v}{[S]} \\ & b = V_{\max } \\ & \text{y-intercept: } v_{\max } \\ & \text{x-intercept: } \frac{V_{\text{max}}}{K_m} \\ & \text{slope: } -K_m \end{aligned} \]
The Lineweaver Burk Plot utilizes the reciprocal values of both substrate concentration and the initial reaction rates to create a linear \(\frac{1}{[S]}\) vs. \(\frac{1}{V_{\max }}\) graph. This graph allows the constants \( V_{\max} \) and \( K_m \) to be easily interpreted through computer-generated regression.
To find \( V_{\max} \), we can take the reciprocal of the y-intercept \(\frac{1}{V_{\max }}\). To find \( K_m \), we can substitute the previously calculated \(\frac{K_m}{V_{\max }}\) value into the slope of the graph, \(\frac{K_m}{V_{\max }}\), and solve for \( K_m \). The calculations for these two parameters are demonstrated below:
\[ \begin{aligned} & V_{\max} = \frac{1}{b} \\ & K_m = m \cdot V_{\max} \end{aligned} \]
In the Eadie-Hofstee Plot, the x-axis represents the ratio of velocity to substrate concentration (\(\frac{v}{[S]}\)) while the y-axis represents the initial velocity \(v\).This graph also allows the constants \( V_{\max} \) and \( K_m \) to be easily interpreted through computer-generated regression.
To find \( V_{\max} \), we can locate the y-intercept of the graph. To find \( K_m \), we can take the absolute value of the slope.
By applying either the Lineweaver Burk Plot or the Eadie-Hofstee Plot to the rate-limiting steps of our L-borneol production pathway, we can experimentally obtain the necessary kinetic constants of \( V_{\max} \) and \( K_m \). Then, these values will be utilized in our chain of Michaelis-Menten equations (Tiwardi, n.d.).
Release: Membrane Diffusion
Through our enzyme kinetics model, we can theoretically determine the final yield of L-borneol, which in turn allows us to know its internal concentration within our minicells. This information is critical for accurately modeling the release of L-borneol from minicells.
The transmembrane diffusion rate depends on factors like distance, diffusant size, and velocity. A key factor is the concentration gradient: a steeper gradient drives faster diffusion, but as the internal concentration decreases, the diffusion rate declines (“Factors affecting the rate of diffusion,” 2024). In typical models, internal and external concentrations are constant. However, in our case, borneol production within minicells adds a second variable that impacts internal concentration and thus the diffusion rate. To accurately capture this dynamic, our model incorporates the ongoing borneol production's effect on the concentration gradient.
Method
The parameters of our model include:
\[ \begin{aligned} & C(t): \text{ Concentration of borneol inside the minicell at time } t \\ & R: \text{ Rate of internal production of borneol} \\ & D: \text{ Diffusion coefficient of borneol} \\ & V: \text{ Volume of the minicell} \\ & A: \text{ Surface area of the minicell} \\ & L: \text{ Thickness of the minicell membrane} \end{aligned} \]
An elementary way of modeling the concentration gradient of borneol across the minicell membrane would be \( \frac{dC}{dx} = \frac{C_{\text{inside}} - C_{\text{outside}}}{L} \) , the difference of the internal and external borneol concentration divided by the membrane thickness, \(L\).
In this scenario, due to the lack of borneol in the environment, we assume \(C_{\text{outside}}\) to be close to zero. With this assumption, our former equation can be modified into: \[ \frac{dC}{dx} \approx \frac{C_{\text{inside}}}{L} \]
Following this, \( \frac{C_{\text{inside}}}{L} \) can be substituted into Fick’s First Law, \(J = -D \frac{dC}{dx}\), to obtain: \[ J = -\frac{D C_{\text{inside}}}{L} \]
The equation can further be modified by adjusting the diffusion flux from the amount of borneol diffused per area per unit time to the amount of substance per area per unit time at time \(t\): \[ J(t) = -\frac{D}{L} C(t) \]
This is then multiplied by \(A\), the area of diffusion, and transformed into \( J(t) \cdot A = -\frac{AD}{L} C(t), \) the rate of the outward diffusion of borneol at time \(t\).
On a closely related subject, another vital model concept is as follows: \[ \text{change in concentration at time } t = \frac{\text{rate of production at time } t + \text{rate of exit at time } t}{\text{minicell volume}} \]
To find the change in the internal concentration of borneol at time \(t\), we need to combine the negative rate at which borneol diffuses out of the minicell and the rate at which borneol is produced within the cell at time \(t\) and divide the sum by minicell volume \(V\).
The rate of borneol production, \(R\), is assumed to be constant throughout the entire lifespan of the minicell. After substituting all of the variables, the following equation is produced: \[ \frac{dC(t)}{dt} = \frac{R - A \cdot D \cdot \frac{C(t)}{L}}{V} \]
The equation above can then be rearranged and simplified to solve for \(C(t)\), the internal concentration of borneol at time \(t\): \[ C(t) = \frac{R \cdot L}{AD} + C_0 \cdot e^{-\frac{AD}{LV} t} \]
The constant of integration, \(C_0\), is assumed to be the initial condition of the minicell at time \(t = 0\). Inputting \(t = 0\) into our most recent equation generates the equation below: \[ C(0) = \frac{R \cdot L}{AD} + C_0 \cdot e^{-\frac{AD}{LV} \cdot 0} = \frac{R \cdot L}{AD} + C_0 \]
After a brief simplification, we get: \[ C(0) = \frac{R \cdot L}{AD} + C_0 \]
This can then be reordered to produce: \[ C_0 = C(0) - \frac{R \cdot L}{AD} \]
And we can substitute \(C_0\) into our former solved function \(C(0)\) and replace the constants \(\frac{AD}{L}\) with \(K\) for the following equations: \[ C(t) = \frac{RL}{AD} + \left(C(0) - \frac{RL}{AD}\right) e^{-\frac{AD}{LV}t} \] \[ C(t) = \frac{R}{K} + (C(0) - \frac{R}{K}) e^{-\frac{K}{V} t} \]
The original equation for the rate of exit for borneol, \(J(t) \cdot A = \frac{AD}{L} C(t) = K \cdot C(t)\), can be updated by substituting \(C(t)\) with our solved equation. We get: \[ \text{rate of exit at time } t = K \cdot \left( \frac{R}{K} + (C(0) - \frac{R}{K}) e^{-\frac{K}{V} t} \right) \]
We then take the integral of this equation, \(\int_0^t K \cdot C(t) \, dt\), to obtain the amount of borneol that diffused out of the minicell after time \(t\) passed: \[ \int_0^t K \cdot C(t) \, dt = R \cdot t - V(C(0) - \frac{R}{K}) e^{-\frac{K}{V} t} + c \]
The constant of integration, \(C\), as we mentioned beforehand, is the initial concentration of borneol outside the minicell. Thus, we have the final equation, which is as follows: \[ \int_0^t K \cdot C(t) \, dt = R \cdot t - V(C(0) - \frac{R}{K}) e^{-\frac{K}{V} t} + C(0) \]
Obtaining Parameters
The diffusion coefficient of borneol is a variable that is widely used within our equations. We determined it through the Stokes-Einstein-Sutherland equation, which is applied for spherical diffusants within liquids (Baer et al., 2023). In our experiment, we assume the geometry of borneol to be spherical, which makes modeling it easier and thus giving us the equation for its diffusion coefficient below, where: \[ D = \frac{k_B T}{6 \pi \mu R} \] with the variables:
- \( k_B \): Boltzmann’s constant
- \( T \): temperature (K)
- \( \mu \): solvent viscosity
- \( R \): solute radius
-
In our model, the temperature will be set at room temperature, averaging around 23℃ or 296.15K, as this will be the average temperature in which our product is used (“The climate of Taiwan,” n.d.).
-
The solvent in our scenario would be the cell membrane, as it is the medium in which our solute, borneol, diffuses through. The viscosity of our minicell membrane was determined to be around 950 centipoise (Mika et al., 2016).
-
The estimation of borneol’s radius was done through the density equation, \( d = \frac{m}{V_b} = \frac{M_r}{n \cdot V_b} \), where the density of borneol is equal to its mass over volume, or more specifically, its molecular mass over its volume multiplied by Avogadro’s number.
Rewriting the equation gives us \( V_b = \frac{M_r}{n \cdot d} \), and \( V_b \), the volume of borneol, can be substituted with the standard equation for calculating the volume of spheres. Again, we make the simplification that borneol’s geometry is spherical. We thus get the equation below: \[ \frac{4}{3} \pi r^3 = \frac{M_r}{n \cdot d} \]
Reordering the equation to solve for \( r \) then gives us our final equation: \[ r = \sqrt[3]{\frac{3 M_r}{4 \pi n \cdot d}} \]
We can substitute the values for the density and molar mass of borneol, both of which have been determined through external research (PubChem, n.d.), to get: \[ r = \sqrt[3]{\frac{3 \times (154.25)}{4 \pi \times (6.022 \times 10^{23}) \times 1.011}} = 3.9 \times 10^{-8} \, \text{cm/molecule} = 0.39 \, \text{nm/molecule} \]
Thus, the radius of our solute, borneol, was approximated to be 0.39 nm.
Table 2. Summary of values needed to find borneol’s diffusion coefficient
Thus with all variables obtained, we can solve for the diffusion coefficient of borneol. Solving for it with the Stokes–Einstein–Sutherland equation, we get \( D = 0.005894 \times 10^{-10} \, \text{m}^2/\text{s} \).
Another parameter needed would be \( V \), the volume of our minicells. As most research describes the size of minicells by their width (diameter) and not their volume, we would have to calculate the value ourselves with the equation: \[ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 \]
Minicell sizes can vary significantly in different scenarios; their diameters have been recorded to range anywhere from 100–400 or 800 nm (Kim et al., 2022; Yu et al., 2021). We chose to simulate the volume of our minicells with the assumed diameter of 800 nm, as proving the efficacy of our product with the most extreme scenario (having the largest solute and therefore the slowest diffusion speed) will mean that our product can function in all other situations.
Inputting \( d = 800 \, \text{nm} \) (or \( r = 400 \, \text{nm} \)) into the equation gives us the approximation for the maximum volume of our minicell: \[ V = \frac{4}{3} \pi \cdot 400^3 \] \[ V \approx 0.268 \, \mu m^3 = 2.68 \times 10^{-19} \, m^3 \]
Additionally, the radius of our minicell can be employed to also calculate each minicell’s surface area. Using the equation \( A = 4 \pi r^2 \), the surface area of an individual minicell is approximately: \[ A = 4 \pi \cdot 400^2 \] \[ A = 2.01 \, \mu m^2 = 2.01 \times 10^{-12} \, m^2 \]
Lastly, the thickness of our minicell membranes, \( L \), is determined to be 7.5 nanometers, or \( 7.5 \times 10^{-9} \, \text{meters} \) (“Thickness of cytoplasmic and outer membrane,” n.d.).
Ultimately, the constant \( K \), which is equal to \( \frac{AD}{L} \), will be \( K = 1.58 \times 10^{-16} \).
Thus, with all parameters obtained, they can be substituted into our model to produce the final version: \[ R \cdot t - \left(2.68 \times 10^{-19}\right)(C(0) - \frac{R}{1.58 \times 10^{-16}}) e^{-589.55t} \]
or
\[ R \cdot t - V(C(0) - \frac{R}{K}) e^{-\frac{K}{V} t} + C(0) \]
where:
- \( R \): Rate of internal production of L-borneol
- \( V \): Volume of the minicell
- \( K \): \( \frac{A \cdot D}{L} \), diffusion constant
- \( D \): Diffusion coefficient of L-borneol
- \( A \): Surface area of the minicell
- \( L \): Thickness of the minicell membrane
- \( C(0) \): Initial internal concentration of L-borneol
Model Simulation
After writing a Python code, we were able to plot the external concentration of L-borneol over time given any initial internal concentration and internal production rate.
Figure 5. The external concentration of borneol over time (placeholder value for initial concentration)
Results & Conclusion
The lack of key kinetic constants from the production model has prevented us from accurately determining the L-borneol production rate. This has also affected the diffusion model from connecting with the production model. In the absence of an accurate L-borneol production rate, the diffusion dynamics of L-borneol cannot be precisely simulated. However, our model demonstrates the relationship between the initial glucose concentration inputted and the amount of L-borneol diffused if further constants are obtained.
Limitations
It is known that the rate of diffusion is affected by L-borneol concentration. As the initial concentration of L-borneol increases, the diffusion coefficient also increases at an exponential rate. This is unintuitive as the coefficient is normally a constant that does not change with concentration; however, L-borneol’s ability to disrupt the lipid bilayers makes it so that increased concentrations increase the diffusion coefficient, and, therefore, the rate of membrane diffusion. Nonetheless, we can make the inference that our product would only be even more effective when the rate of diffusion increases together with L-borneol concentration.
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