SRB are widely distributed in sewers and produce large amounts of H2S gas, which poses a safety hazard, so trying to inhibit the growth of SRBs and their biofilms using E. coli requires modeling both bacteria at the cellular community level. In this part, we combined some of the results obtained at the gene level model in the previous step with the Constraint-based reconstruction and analysis (COBRA) model and the Agent-based Model (ABM) model for co-culture simulations to verify that the addition of E.coli can have an effect on the growth of SRB and its biofilm in the sewers. SRB and its biofilm growth were effectively inhibited by the addition of E.coli.
Cell Layer Flowchart
Our GSM model for E.coliis built based on the BiGG platform, and the GSM model for SRB was obtained from https://www.vmh.life. Based on the literature related to macrogenomic analysis of SRB in sewers (Xuan Shi et al., 2023), we adapted the bound of several reactions in the model to simulate the anaerobic state and anaerobic condition and environment in the sewers.
According to the literature (Xuan Shi et al. 2023), the main carbon sources associated with the growth of SRB in sewers are L-lactate, Acetate, Propionate, and the main electron acceptor is Sulfate (SO42-), and we set the concentration of these compounds in the environment to be constant as mentioned in the assumption, and the rest of the inorganic salts and amino acids were calculated by Genome scale metabolic model to simulate the growth environment of the two bacteria.
In biofilms, the diffusion of compounds follows Fick's law, in the surface and shallow layers of biofilms, compounds diffuse more easily and have higher concentrations due to their proximity to the water column. As the thickness of the biofilm gradually increases, the concentration of compounds in the deeper layers is relatively low. This process can be represented by the following diffusion equation:
\begin{equation} C_{m}=C_{m1}-(C_{m1}-C_{m0})·erf(\frac{x}{2\sqrt[]{Dt} } )\tag{1} \end{equation}
Where Cm represents the concentration of compound m at a location, Cm1 represents the highest concentration of compound m over a distance, Cm0 represents the lowest concentration of compound m over a distance, x represents the distance from the location to the location of the highest concentration, D represents the diffusion coefficient of the compound, t represents the time, and erf(β) is the Gaussian Error Function, which has the following equation:
\begin{equation} erf(\beta )=\frac{2}{\sqrt[]{\pi} }\int_{0}^{\beta }exp(-\beta ^2)d\beta \tag{2} \end{equation}
so long as
\begin{equation} \beta =\frac{x}{2\sqrt[]{Dt} } \tag{3} \end{equation}
The diffusion coefficient in biofilms, on the other hand, can be defined by equation (4) (B. D'Acunto et al. 2011):
\begin{equation} D=7.4 \times 10^{-8} \; \frac{\left(\varphi_{b} M_{b}\right)^{0.5} T}{\mu_{b} V_{a}^{0.6}} \tag{4} \end{equation}
where ϕb is the solvent association parameter (2.6 for water), Mb is the molecular weight of the solvent (18 g for water), T is the absolute temperature (about 300 K in the sewer). μb is the solvent absolute viscosity (0.8360 cp for water at 26.85 °C, 300K), Va are the molecular volumes of the solute as liquid at its normal boiling point (cm³ mol-1). And the diffusion coefficients in water were multiplied by a factor of 0.8 to correct the additional diffusion resistance in the biofilm.
The concentration of the compound at the corresponding depth in the biofilm can be found by calculating the above equation.
Flux-Balance Analysis (FBA) is one of the effective tools in microbiology for quantitatively analyzing the fluxes of cellular reactions, and sometimes it is only necessary to know the corresponding fluxes of a few reactions to find the fluxes of the rest of the reactions in the whole cellular reaction network by FBA. In the hybrid model of E.coli and SRB, we only need to get the corresponding fluxes of their uptake of carbon sources (or electron acceptors) to find the fluxes of their biomass growth, thus simulating the growth of bacteria.
Agent-Based Model (ABM) is uniquely suited to simulate biomes as well as infectious disease models, and is widely popular due to its excellent discrete, contextual situated properties. Here, we used ABM to simulate the growth of bacteria and biofilm for 100 days.
In this model, we diffuse the compounds into the metacells of the biofilm and their concentration decreases with depth by using the diffusion equation (1) described above, which will make the bacteria in the deeper layers of the biofilm will not be able to get enough carbon source to die. In contrast, in the metacells where the carbon source is more abundant, we used Michaelis equations for the entry of the corresponding compounds into the cell to quantify the flux of bacterial responses in each metacell:
\begin{equation} V_{B,m}=\frac{V_{max}·S_{m}}{K_m+S_m}·\frac{Biomass_B·\Delta x^3}{W_B}\tag{5} \end{equation}
where vB,m is the rate of uptake of compound m by Bacteria B in a given metacell (mM-h-1), Vmax is the maximum rate of uptake of the compound (mM-h-1), Sm is the substrate concentration (mM), which is obtained from the biofilm thickness and diffusion equations, Km is the Mie constant of the consumption of the compound (mM), BiomassB is the biomass of Bacteria B in the metacell (g-L- 1), Δx³ is the volume of the metacell (10-9 L), and wB is the weight of a single bacterium of Bacterium B (1.4 × 10-12 g for E.coli, 6 × 10-12 g for SRB). In this equation, we can quantify the rate of consumption of a compound by bacteria within a single metazoan. Subsequently, we can get the reaction flux of that compound into the cell in the GEM model by transformation:
\begin{equation} bound_{B,m}=\frac{V_{B,m}}{\delta } \tag{6} \end{equation}
where boundB,m is the flux of the uptake reaction of bacterial B in a given metacell (mmol-gDw-1-h-1) and σ is the dry weight-to-volume ratio of bacterial cells (300 gDw-L-1, SCU-CHINA, 2022). The obtained bounds can be imported into COBRA to calculate the growth of bacterial biomass:
\begin{equation} v_{B,biomass}=FBA(B,biomass)\tag{7} \end{equation}
where vB,biomass is the rate of biomass growth (g-h-1) of Bacteria B in a given metazoan, and FBA(B, biomass) is the result of the calculated response flux of FBA biomass growth of Bacteria B in that metazoan. Subsequently, we can constrain the growth of both bacteria by logistic equation and introduce an environmental tolerance K (15 g-L-1 for SRB, 12 g-L-1 for E.coli):
\begin{equation} \Delta Biomass_B=v_{B,biomass}·(\frac{1}{1+exp(Biomass_B-K_B)})·\Delta t \tag{8} \end{equation}
where ΔBiomassB is the biomass growth of bacteria B and Δt is the unit time. Additionally, since E. coli produces antimicrobial peptides which kill SRB, and the environmental AHL concentration also affects how much ccdA protein E. coli produces, which determines its life and death, the mortality formula for both is introduced:
\begin{equation} Death_{SRB}=d_{SRB}+(1-d_{SRB})(\frac{1}{1+exp(\theta (Biomass_{ECO}-K_{ECO}))})\tag{9} \end{equation}
\begin{equation} Death_{ECO}=d_{ECO}+(1-d_{ECO})(\frac{\phi _1}{\phi _2+exp(\phi _3 (\log _{10}(AHL)-\phi _4))})\tag{10} \end{equation}
where Death denotes the mortality rate of the two bacteria, d denotes the base mortality rate of the two bacteria, DeathSRB is determined by the logistic equation for the concentration of E.coli, while DeathECO is determined by the logistic equation for the concentration of AHL, where φi denotes the individual parameters of the logistic equation, and integrating Eqs. (8,9,10), one can obtain the updated equation for Biomass:
\begin{equation} Biomass_B^+=(Biomass_B+\Delta Biomass_B)\times (1-Death_B)\tag{11} \end{equation}
In our ABM, in addition to the calculation of biomass in individual metazoans, the growth, diffusion and interaction of bacterial communities and biofilms in three-dimensional space must also be taken into account. Therefore, we used a three-dimensional Cellular Automata (CA) to simulate the bacterial community level. In the 3D CA, we employed von Neumann-type neighborhoods (6-neighborhoods) to diffuse E. coli, SRBs, AHLs, and biofilms using a concentration gradient equation:
\begin{equation} M_{biofilm}(i,j,k)=\alpha M_{biofilm}(x,y,z)+\beta M_{AHL}(i,j,k)\tag{12} \end{equation}
\begin{equation} M_{AHL}(i,j,k)=r_{AHL}(M_{AHL}(x,y,z)-M_{AHL}(i,j,k))\tag{13} \end{equation}
\begin{equation} M_{SRB}(i,j,k)=r_{SRB}(M_{SRB}(x,y,z)-M_{SRB}(i,j,k))\tag{14} \end{equation}
\begin{equation} M_{ECO}(i,j,k)=r_{ECO}(M_{ECO}(x,y,z)-M_{ECO}(i,j,k))\tag{15} \end{equation}
where Mx is each matrix, (x,y,z) represents the coordinates of a certain metacell, (i,j,k) is some neighborhood of (x,y,z), α is the diffusion coefficient between biofilms, β is the coefficient of the influence of AHL on biofilm formation, and rx is the diffusion coefficient of each matrix. With Eqs. 12-15, we can discretely update each substance matrix, thus realizing the update and iteration of the metacellular automaton.
Fig.1 Compound concentrations at different biofilm thicknesses
As shown in the figure, after the calculation of the diffusion equation, we can easily obtain the corresponding concentrations of several compounds from the surface of the biofilm to the interior 2000 μm, which will be used as inputs for the calculation of the fluxes of the relevant compounds utilized by the bacteria in the COBRA model.
In the biofilm growth section of the results, we performed a visualization of biofilm growth, and in that section, we will give a plot of the change in concentration of the four compounds with biofilm growth based on biofilm growth:
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Fig.2 Plot of acetic acid changes (left: 3d, right: 2d)
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Fig.3 L-lactic acid changes (left: 3d, right: 2d)
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Fig.4 Propionic acid changes (left: 3d, right: 2d)
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Fig.5 Sulfate changes (left: 3d, right: 2d)
Fig.6 Biofilm thickness reference (Shi, Xuan et al. 2020)
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Fig.7 Comparison of biofilm formation (left: SRB alone, right: addition of E.coli and SRB mixed culture)
As shown in the figure, when SRBs were grown alone, the biofilm formation was very rapid and comparable to the biofilm thicknesses in the literature (Shi, Xuan et al. 2020) at different times; whereas, when we randomly added a small amount of E.coli at the bottom at the initial time of the biofilm formation, we could make the biofilm growth rate to be limited.
Fig.8 Biofilm thickness over time
As shown, in our model, the thickness of biofilm will significantly decrease with the addition of E. coli. In the simulation of 100 days of biofilm growth, the biofilm thickness will reach 2000 μm when only SRB are present in the environment, while the biofilm thickness grows slowly and stays around 600 μm with the addition of the engineered bacteria co-cultured with SRB.
Fig.9 Bacterial weight over time
As shown in the figure, the significant reduction of biofilm thickness in our model will make the SRB bacterial weight decrease as well, and in the 100-day simulation, the SRB bacterial weight after co-cultivation of the engineered bacteria with SRB finally stays at 0.399 g, whereas the bacterial weight of the SRB-only condition will eventually reach 2.64 g. This means that our engineered bacteria can not only effectively reduce the biofilm production, but also effectively reduce the SRB abundance and realize the inhibition of SRB growth.
| Parameters | Value | Explanations |
|---|---|---|
| \(\Delta x\) | \(10^{-4}\) | Length of each metacell (m) |
| \(\Delta t\) | 24 | Iteration interval (h) |
| \(W_{SRB}\) | \(6·10^{-12}\) | Individual SRB weight (g) |
| \(W_{ECO}\) | \(1.4·10^{-12}\) | Individual E.coli weight (g) |
| \(C_{ace,1}\) | 0.666 | Initial concentration of acetate (mM) |
| \(C_{ppa,1}\) | 0.391 | Initial concentration of propionate (mM) |
| \(C_{lac,1}\) | 0.370 | Initial concentration of L-lactate (mM) |
| \(C_{so4,1}\) | 0.260 | Initial concentration of sulfate (mM) |
| \(\delta\) | 300 | Ratio of dry weight by volume (gDw·L-1) |
| \(K_{SRB}\) | 15 | Environmental capacity of SRB (g·L-1) |
| \(K_{ECO}\) | 12 | Environmental capacity of E.coli (g·L-1) |
| \(d_{SRB}\) | 0.2 | Base mortality rate of SRB |
| \(d_{ECO}\) | 0.1 | Base mortality rate of E.coli |
| \(\theta\) | -0.5 | Parameters for calculating mortality in SRB |
| \(\phi _1\) | 1.503 | Parameters 1 for calculating mortality in E.coli, determined by the gene layer model |
| \(\phi _2\) | 1.5072 | Parameters 2 for calculating mortality in E.coli, determined by the gene layer model |
| \(\phi _3\) | 4.6672 | Parameters 3 for calculating mortality in E.coli, determined by the gene layer model |
| \(\phi _4\) | 0.1501 | Parameters 4 for calculating mortality in E.coli, determined by the gene layer model |
| \(\alpha\) | \(10^{-2}\) | Biofilm Matrix Update Parameter 1 |
| \(\beta\) | \(10^{-4}\) | Biofilm Matrix Update Parameter 2 |
| \(r_{AHL}\) | 0.1 | AHL Matrix diffusion Parameter |
| \(r_{SRB}\) | \(U(0, 0.1)\) | SRB matrix diffusion parameters, U(a, b) represents the uniform distribution of the interval (a, b) |
| \(r_{ECO}\) | \(U(0, 0.1)\) | E.coli matrix diffusion parameters, U(a, b) represents the uniform distribution of the interval (a, b) |
Table 1. All parameters used above, explanations and corresponding references
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