Intracellular Propionic Acid Synthesis Model

The following section explains how E. coli gassle produce propionic acid in response to nitric oxide (NO) produced at inflammatory sites. We introduced the PnorV system to enable E. coli gassle to detect NO produced at inflammatory sites in the gut. In this system, NO binds to the transcriptional regulator NorR, which binds upstream of the promoter to activate transcription.

Figure 3. Mechanism of the PnorV system. NorR forms a homo hexamer and binds to an enhancer sequence upstream of PnorV. The binding of heme iron to NorR activates the transcription of downstream genes.

We also introduced the Sleeping beauty mutase (Sbm) operon downstream of PnorV. It encodes an enzyme necessary for the synthesis of propionic acid from Succinyl-CoA and Succinate in the TCA cycle in E. coli.

Figure 4. Central metabolic pathway of E. coli and Sbm pathway.The pathway with a skyblue background of the figure is the Sbm pathway.

In summary, the combination of Succinyl-CoA and Succinate biosynthesized in the central metabolic pathway, the PnorV system, and the Sbm operon enables propionic acid synthesis in response to NO. Since the sequence of these steps is complex, we simplified it and modeled the process of propionic acid synthesis (Figure 5).

The assumptions made in the model are as follows.

• The expression levels of each gene in the Sbm operon are the same. That's because the ribosome binding sites (RBSs)upstream of each gene in the Sbm operon are different, it is difficult to estimate the exact expression levels.)
• kcat/Km of each protein (Sbm, YgfD, YgfG, and YgfH) encoded by the Sbm operon was compared and found to be the lowest in the reaction catalyzed by the Sbm protein, which was determined to be the rate-limiting step.
• The volume and surface area of E. coli gassle do not change during the cell growth process. Therefore, the dilution effect of the concentration of intracellular substances is not considered.
• The volume of an E. coli gassle is 1 µm³, the surface area is 10 µm², and the thickness from the inner membrane to the outer membrane including the periplasm is 20 µm.
• It is assumed that introducing the Sbm operon did not cause a significant change in the intracellular Succinyl-CoA concentration.

Figure 5. Intracellular model of propionic acid synthesis. NorR has two states: NO-free NorR_r and NO-bound NorR_a. The former inhibits promoter activity, while the latter activates it.

Based on Figure 5, the intracellular nitric oxide concentration (NO_in), NorR protein concentration (NorR), Sbm protein concentration (Sbm), intracellular propionate concentration (Pro_in), and intestinal propionic acid can be described as follows using differential equations.

Equation for Intracellular Propionate Synthesis Model

Model of Intestinal Epithelial Permeation of Propionic Acid

Propionic acid produced by E. coli gassle is released into the intestine. Some of them permeate the intestinal epithelium. To account for this effect, this project used a compartment model.

Figure 6. Image of intestinal epithelial permeation of propionic acid.

Considering the membrane permeability coefficient (Peff), the effective surface area (SA), and the volume (V) of the human colon concerning the phenomenon in Figure 6, the mass of propionic acid (A) absorbed by the intestinal epithelium is described by differential equations.

Equation for Intestinal Epithelial Permeabilization Model

\[ \frac{d\mathit{A}}{dt} = P_{eff} \cdot SA \cdot \mathit{C} \]

Intestinal Gas Release Model

Intestinal gas is released as farts. Therefore, the reduction of propionic acid in intestinal gas due to farting (B) was described.

Figure 7. Image of intestinal gas emission.

The assumptions made in the model are as follows.

• The volume of a fart is the median fart an adult makes per fart (v).
• Farts occur 14 times a day [11] and at equal intervals.

Equation for Intestinal Gas Emission Model

\[ \frac{d\mathit{B}}{dt} = v \cdot \mathit{C} \cdot \sum_{l=0}^{\infty} \delta(t - 103 \cdot l) \]

l is an integer and δ refers to the Dirac delta function.

Overall Model

The intracellular propionic acid synthesis model, the intestinal epithelial permeation model for propionic acid, and the intestinal gas release model were combined to describe the intestinal propionic acid concentration (C).

The assumption made in the models is shown below.

• The propionic acid produced in the intestinal tract is free of other factors that may reduce propionic acid, such as degradation by other intestinal microbiota.

Equations for Intestinal Propionic Acid Model

\[ \frac{d[\mathit{NO}_{in}]}{dt} = a_{NO} \cdot NO_{out} - b_{NO} \cdot [\mathit{NO}_{in}] \]

\[ \frac{d[\mathit{NorR}]}{dt} = a_{NorR} - b_{NorR} \cdot [\mathit{NorR}] \]

\[ \frac{d[\mathit{Sbm}]}{dt} = a_{Sbm} \cdot \frac{K_r}{K_r + [\mathit{NorR}]} \cdot \frac{[\mathit{NorR}]}{[\mathit{NorR}] + K_A} \cdot \frac{[\mathit{NO}_{in}]^n}{[\mathit{NO}_{in}]^n + K_a} - b_{Sbm} \cdot [\mathit{Sbm}] \]

\[ \frac{d[\mathit{Pro}_{in}]}{dt} = \frac{k_1 \cdot [\mathit{Sbm}] \cdot [SuccinylCoA]}{K_m + [SuccinylCoA]} - b_{Pro_{in}} \cdot [\mathit{Pro}_{in}] - a_{Pro_{out}} \cdot [\mathit{Pro}_{in}] \]

\[ \frac{d\mathit{C}}{dt} = a_{Pro_{out}} \cdot [\mathit{Pro}_{in}] \cdot M_{pro} \cdot N - v \cdot \mathit{C} \cdot \sum_{l=0}^{\infty} \delta(t - 103 \cdot l) - \frac{1}{V} \cdot P_{eff} \cdot SA \cdot \mathit{C} \]

\[ r = \frac{\mathit{C}}{m} \]

Intestinal Propionic Acid Model Tables

Variables Table

Variable	Description	Units
NOin	Intracellular nitric oxide concentration	nM
NorR	NorR concentration	nM
Sbm	Sbm concentration	nM
Proin	Propionic acid concentration in E. coli	nM
C	Intestinal propionic acid concentration	g/L
r	Intestinal propionic acid concentration	ppm

Parameters Table

Parameter	Description	Value	Units	Reference
aNO	NO uptake rate constant	4.2	1/min	1
bNO	Degradation rate of NO	4.15e2	1/min	2
aNorR	Combined transcription/translation rate of NorR	3	nM/min	3
bNorR	Degradation rate of NorR protein	0.3	1/min	3
aSbm	Combined transcription/translation rate of Sbm	200	nM/min	3
Kr	Binding affinity between non-induced, repressed NorRr and PnorV	0.1	nM	3
KA	Binding affinity between induced, activated NorRa and PnorV	5	nM	3
NOout	Nitric monoxide concentration	7.3, 1255	nM	4
n	Hill coefficient for NO-NorR binding	2	dimensionless quantity	3
Ka	Binding affinity between NO and NorR	10	nM	3
bSbm	Degradation rate of Sbm protein	0.3	1/min	5
k1	Propionic acid synthesis rate constant	12	1/min	6
Km	Michaelis constant of Sbm and Succinyl-CoA	11.2e3	nM	6
bProin	Propionic acid degradation rate constant	5.907e-13	1/min	7
aProout	Propionate transport constant into the intestinal tract	4.2	1/min	1
SuccinylCoA	Succinyl-CoA concentration	2.3e5	nM	8
SA	Colonic Intestinal Surface Area	2.0e4	cm2	9
Peff	Effective permeability of propionate	4.2e-3	cm/min	10
v	Fart volume	90	ml	11
V	Volume of large intestinal tract	1450	cm3	12
m	Mass of air in standard condition	1.286e3	g/L	13
Mpro	Molecular weight of propionic acid	74e-9	g/nmol	14
N	E. coli cell count	1.4e5~2.1e8	cells	Estimated

When these parameters were incorporated into the model and simulated, the following graph was obtained.

A) N = 1.4 × 10⁵, NO_out = 7.3

B) N = 1.4 × 10⁵, NO_out = 1255

C) N = 2.0 × 10⁸, NO_out = 7.3

D) N = 2.0 × 10⁸, NO_out = 1255

E) N = 2.1 × 10⁸, NO_out = 7.3

F) N = 2.1 × 10⁸, NO_out = 1255

Figure 8. Simulation of the Propionic Acid Model in the Intestinal Tract. A) N = 1.4 × 10⁵, NO_out = 7.3, B) N = 1.4 × 10⁵, NO_out = 1255, C) N = 2.0 × 10⁸, NO_out = 7.3, D) N = 2.0 × 10⁸, NO_out = 1255, E) N = 2.1 × 10⁸, NO_out = 7.3, F) N = 2.1 × 10⁸, NO_out = 1255.

The human olfactory threshold for propionic acid is 0.0057 ppm [25]. Assuming that flatulence reaches a person’s nose while maintaining the propionic acid concentration from the intestinal tract, it is estimated that if there are around 1.4 × 10⁵ to 2.0 × 10⁸ gassle in the gut, the propionic acid concentration would be high enough to be detectable by smell. This could allow for distinguishing between healthy individuals and patients based on the odor of their flatulence.

However, if there are more than 2.1 × 10⁸ gassle in the gut, the propionic acid concentration in healthy individuals would reach the human olfactory threshold. As a result, it would no longer be possible to distinguish between healthy individuals and patients based on the smell of their flatulence.

Furthermore, assuming that recombinant organisms do not significantly proliferate in the gut, it is estimated that approximately 1.0 × 10⁷ gassle would need to be encapsulated in a drug capsule. Based on this conclusion, the concentration of gassle in the capsule and the size of the capsule can be determined.

Refer to the Hardware page.

HSV-TK/GCV System Model

Detailed description of the HSV-TK/GCV System Model, including its design and functional aspects.

In our project, we introduce gassle into the body as probiotics. Therefore, it is necessary to ensure that gassle is not released into the environment by killing them at any desired time. To achieve this, we utilized the HSV-TK/GCV system. This system employs GCV, a deoxyguanosine (dG) analog.

Figure 9. Structures of GCV and dGTP. These molecules share a highly similar structure.

After the phosphorylation of GCV by HSV-TK, Ganciclovir triphosphate (GCVTP) is synthesized through further phosphorylation by intracellular kinases. GCVTP competitively inhibits the incorporation of dGTP into the double-stranded DNA, thereby preventing DNA elongation. As a result, the proliferation of Escherichia coli gassle is inhibited.

Inhibition of Escherichia coli Gassle Proliferation by Ganciclovir (GCV)

Following the phosphorylation of GCV by HSV-TK, Ganciclovir triphosphate (GCVTP) is synthesized through additional phosphorylation by intracellular kinases. GCVTP competitively inhibits the incorporation of dGTP into double-stranded DNA, thereby blocking DNA elongation and ultimately inhibiting the proliferation of Escherichia coli gassle.

Assumptions Made in the Model:

• The volume and surface area of Escherichia coli gassle were assumed to remain constant throughout the cell growth process. Therefore, the dilution effect of intracellular substance concentrations was not considered.
• The volume of Escherichia coli gassle was assumed to be 1 µm³, the surface area 10 µm², and the thickness from the inner membrane to the outer membrane, including the periplasm, was set at 20 µm.
• Phosphorylation from GCV to GCVTP occurs through multiple steps, but the reaction catalyzed by HSV-TK was assumed to be the rate-limiting step.
• Since a constitutive promoter is located upstream of the HSV-TK gene, the expression level of HSV-TK was assumed to remain constant over time.
• As the expression level of HSV-TK could not be quantified in this project, it was roughly estimated by dividing the number of intracellular protein molecules [20, 23] by the total number of protein types in Escherichia coli gassle (approximately 4,000 types) [24].
• The extracellular GCV concentration was assumed to remain unchanged over short periods.
• The impact of GCV uptake on intracellular dGTP concentration was not considered.
• The environmental carrying capacity was assumed to be about 0.1% of the total number of gut bacteria in the human intestine.
• The apparent cell doubling time of gassle upon GCV arrival in the intestinal tract was assumed to be approximately 40 minutes [21].
• The number of Escherichia coli gassle cells at the moment GCV reached the intestinal tract was assumed to be 1.4 × 10⁶, a number that can distinguish between healthy individuals and patients.

Figure 10. HSV-TK/GCV System Model Diagram.

The time-dependent changes in intracellular GCV concentration (GCV_in), GCVTP concentration (GCVTP), and the number of Escherichia coli gassle cells in the intestinal tract (N) were described using differential equations. Although the value of k1 was initially planned to be determined by fitting the cell growth measurement results using the least squares method, the desired results were not obtained. Therefore, k1 was set to 0.8.

Equation for HSV-TK/GCV System Model

\[ \frac{d[\mathit{GCV_{in}}]}{dt} = a_{gcv} \cdot (GCV_{out} - [\mathit{GCV_{in}}]) - \frac{k_{cat_p} \cdot HSVTK \cdot [\mathit{GCV_{in}}]}{K_{m_{gcv_p}} + [\mathit{GCV_{in}}]} - b_{gcv} \cdot [\mathit{GCV_{in}}] \]

\[ \frac{d[\mathit{GCVTP}]}{dt} = \frac{k_{cat_p} \cdot HSVTK \cdot [\mathit{GCV_{in}}]}{K_{m_{gcv_p}} + [\mathit{GCV_{in}}]} - b_{gcvtp} \cdot [\mathit{GCVTP}] \]

\[ \frac{d\mathit{N}}{dt} = r \cdot \mathit{N} \cdot \frac{K - \mathit{N}}{K} \left(1 - k_1 \cdot \frac{[\mathit{GCVTP}]}{[\mathit{GCVTP}] + [\mathit{dGTP}]}\right) - d \cdot \mathit{N} \]

Variables Table

Variable	Description	Units
GCVin	Intracellular ganciclovir concentration	mM
GCVTP	Ganciclovir triphosphate	mM
N	E. coli cell count	cells

Parameters Table

Parameter	Description	Value	Units	Reference
aGCV	Ganciclovir uptake rate constant	0.22	1/min	15, 16
GCVout	Extracellular GCV concentration	0~1	mM	Estimated
kcatp	The turnover numbers for SR39	0.21	1/min	17
Kmgcvp	The Km values (GCV) for mutant SR39	3.33e-3	mM	18
HSVTK	SR39 protein concentration	1.568e-2	mM	20, 23
bgcv	Degradation rate of GCV	3.21e-3	1/min	19
bgcvtp	Degradation rate of GCVTP	3.21e-3	1/min	19
r	Internal natural growth rate	0.025	1/min	20
K	Environment capacity	1.0e11	cells	22
k1	Rate constant for decrease by GCVTP	0.8	1/min	From wet Data
dGTP	dGTP concentration	6.2e-2	mM	24
d	Internal natural death rate	0.0248725	1/min	21

When these parameters were incorporated into the model and simulated, the following graph was obtained.

A) GCV_out = 0 mM

B) GCV_out = 1.0 × 10^-6 mM

C) GCV_out = 1.0 × 10^-5 mM

D) GCV_out = 1.0 × 10^-3 mM

E) GCV_out = 1.0 × 10^-2 mM

F) GCV_out = 1.0 × 10^-1 mM

Figure 11. A) GCV_out = 0 mM, B) GCV_out = 1.0 × 10^-6 mM, C) GCV_out = 1.0 × 10^-5 mM, D) GCV_out = 1.0 × 10^-3 mM, E) GCV_out = 1.0 × 10^-2 mM, F) GCV_out = 1.0 × 10^-1 mM.

If Td is defined as the time when the number of gassle cells becomes less than 1, the Td values for each case are as follows:

• A) Does not fall below 1,
• B) Does not fall below 1,
• C) Td ≈ 270,000,
• D) Td ≈ 1,700,
• E) Td ≈ 900.
• E) Td ≈ 800.

Therefore, when k1 = 0.8 and GCV_out = 1.0 × 10^-2 mM, it is estimated that all gassle cells can be eliminated within approximately 15 hours after GCV reaches the intestinal tract. Thus, an intestinal GCV concentration of around 1.0 × 10^-2 mM is considered appropriate.

Prospects

Future directions and potential developments based on the current models and findings.

Source Code

The code used in the Model simulation can be obtained at the following link: GitLab.

References

List of all the references and sources cited throughout the project.

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