CONSORTIUM MODEL

Overview

Our idea to coculture a fermentative Saccharomyces. cerevisiae along with Komagataeibacter. sucrofermentans, primarily stems from its ability to produce ethanol, which, when present at specific concentrations, has been shown to improve BC yield^1, ². Furthermore, as yeast is a well-characterised model organism³, it provides us with the opportunity for future optimisations to improve the performance of the consortium. Additionally, using S. cerevisiae would give us the ability to use wider range of substrates to further avoid competition within the consortium.

To better understand the complex interactions of yeast and bacteria in the consortium, we built a mathematical model, based on ordinarily differential equations, following the work of previously established models⁴. The model was then optimised to fit the laboratory data and was used to predict the consortium’s behaviour at different initial conditions. This model would help us assess the overall stability and competition for the substrates in the system

In this way, the model would then be used to identify optimal conditions, such as concentration of carbon source and ethanol in growth media, and inoculation concentrations of yeast and bacteria for the maximum production of bacterial cellulose.

Metabolic Network

The mathematical model for the monocultures of S. cerevisiae and K. sucrofermentans was designed based on a simplified metabolic network, incorporating only the most relevant metabolites and pathways⁴. Based on the literature and our lab experiments, it was observed that the growth of S. cerevisiae was much faster than that of K. sucrofermentans⁵. A cross-feeding mechanism was designed that incorporates ethanol supplementation to establish microbial dependency between S. cerevisiae and K. sucrofermentans, and the fermentative yeast was modelled to grow only on maltose, thereby eliminating competition for the substrates.

The model was designed such that, the initial substrates and final products could be measured, these measurements would then be used to optimise the mathematical model to better suit laboratory data. This process would result in the identification of optimal growth and reaction parameters. Using these parameters, the model would be used to predict the ideal conditions required for maximum cellulose production.

Leveraging various metabolic pathways, we defined the consortium in a way, that K. sucrofermentans would grow predominantly on glucose, and S. cerevisiae would grow predominantly on maltose. K. sucrofermentans would use the glucose to produce cellulose, and for the formation of acetyl-CoA, which acts as a precursor for growth and acetate formation. S. cerevisiae would use maltose for the production of acetyl-CoA, which is a precursor for biomass and ethanol production.

Model

To computationally simulate the consortium, we built an ODE based model, where irreversible Michaelis-Menten (MM) kinetics were used to represent, the change over time and the consumption of the various state variables. The Michaelis-Menten equations, describing the rates of various reactions are reported below (Table 2).

d[P] dt = V_max × [S] K_s + [S]

Where P is the concentration of the product (mmol L^-1),

S is the concentration of the substrate (mmol L^-1),

K_m is the Michaelis constant (mmol L^-1), and

V_max is the maximum reaction rate (mmol L^-1 h^-1).

The Michaelis–Menten constant or K_m represents the substrate concentration at which the rate of the reaction is half the maximal rate. K_m is also indicative of enzyme affinity — a higher value implies a lower binding affinity (meaning a larger concentration of substrate is required for the reaction to occur), while a lower value implies a higher binding affinity (meaning a smaller concentration of substrate is sufficient). The term V_max indicates how fast the reaction can theoretically occur.

The irreversible MM kinetics equation was used to describe most of the steps within the metabolic framework. The only exceptions were the diffusion reactions, describing the transport of ethanol and acetate across the cell membrane, which were written as follows:

d[S’] dt = D × (S − S’)

where S is the concentration of the compound (mmol L⁻¹), S’ is the concentration of the compound across the cell membrane (mmol L⁻¹), and D is the diffusion coefficient (constant) used to describe the rate of diffusion across a permeable membrane. This equation describes the diffusion of a compound across the membrane, based on the concentration gradient.

Growth of the two microbial species were defined using Monod’s equation, which is mathematically similar to irreversible MM kinetics.

μ = μ_max × [S] K_s + [S]

where X is the biomass present (g dry weight), S is the concentration of the substrate limiting the growth (mmol L⁻¹), K_s is the half-saturation constant (mmol L⁻¹), and μ_max is the maximal growth rate (mmol L⁻¹ h⁻¹).

We first set up the mathematical model for the monoculture of S. cerevisiae and K. sucrofermentans, where each step in the metabolic network was represented as an ODE. The initial iteration of the model represented the microbes growing on media where maltose and glucose was the sole carbon source, respectively. During this stage, the kinetic parameters (K_max, and V_max) for the reactions were still unknown. We then ran multiple simulations for the models with the kinetic parameter values chosen randomly between an upper and lower bound, set based on parameter ranges that were biologically possible.

We defined a Sum of Squared Errors (SSE) scoring function to fit the mathematical simulations to the laboratory data; the scoring function is written as:

Ω(p_s) = ∑_i=1^N ( B̂(t_i; p_s) − B_i^data )²

Ω(p_s) represents the scoring (objective) function used to quantify the deviation between model predictions and experimental data. p_s represents the vector of model parameters (e.g., V_max, K_m, diffusion coefficients, growth constants, etc.). N represents the total number of experimental data points. t_i represents the time (or condition) corresponding to the i-th measurement. B̂(t_i; p_s) represents the model-predicted value of variable B (Biomass/OD) at time t_i using parameters p_s. B_i^data represents the experimentally observed value of B at the same time point t_i.

Using this scoring function, we were able to identify the optimal kinetic parameters, which mathematically best describe the growth characteristics observed in the laboratory. These optimised parameters, are reported in the table found in the results section (Table 4).

To model the diffusion of Ethanol and uptake of acetate by S.cerevisiae, we set an upper and lower bound to the diffusion coefficient, and for the kinetic parameters involved in the metabolism of acetate. multiple simulations were produced and the scoring function was used to fit the simulations to the laboratory data obtained by growing S. cerevisiae on medium containing acetate. We followed the similar procedure to account for the uptake of Ethanol and diffusion of acetate by K. sucrofermentans.

The state variables involved in the model, are defined below (Table 1)

List of state variables and their description

Table 1:List of state variables used in the system, along with their descriptions.

Variable	Description
Bacteria
glu_e	external glucose
glu_b	Glucose in bacteria
glc_b	Gluconate in bacteria
glc_e	external gluconate
cll	Cellulose
accoa_b	AcetylCoA in bacteria
et_b	Ethanol in bacteria
ace_b	Acetate in bacteria
et_e	External Ethanol
ace_e	External acetate
Xb	Bacteria biomass
Yeast
mlt_e	External maltose
mlt_y	Internal maltose
ace_y	Acetate yeast
accoa_y	Acetyl-CoA yeast
et_y	Ethanol yeast
Xy	Yeast biomass

The individual reaction rates for various steps in the pathway are displayed in the table below.

Individual reaction rates for various steps in the pathway

Table 2: The individual rates of each reaction within the metabolic network of K. sucrofermentans and S. cerevisiae. The reaction rates are written following irreversible Michaelis-Menten equations for enzymatic processes, and using diffusion equations for the passive diffusion of metabolites.

Equation	Description
Bacteria
$K_{1} = \frac{V_{max, glut}^{b} \cdot [{glu}_{e}]}{K_{m, bglut} + [{glu}_{e}]}$	Glucose transport
$K_{2} = \frac{V_{max, cll} \cdot [{glu}_{b}]}{K_{m, cll} + [{glu}_{b}]}$	Cellulose formation
$K_{3} = \frac{V_{max, glc} \cdot [{glu}_{b}]}{K_{m, glc} + [{glu}_{b}]}$	Gluconate formation
$K_{4} = \frac{V_{max, accoa} \cdot [{glu}_{b}]}{K_{m, accoa} + [{glu}_{b}]}$	Acetyl-CoA formation
$K_{5} = \frac{U_{m}^{b} \cdot [{accoa}_{b}]}{K_{s, accoa} + [{accoa}_{b}]}$	Growth
$K_{6} = \frac{V_{max, et} \cdot [{et}_{b}]}{K_{m, et} + [{et}_{b}]}$	Ethanol consumption $\to$ AcCoA
$K_{7} = \frac{V_{max, aceb} \cdot [{accoa}_{b}]}{K_{m, aceb} + [{accoa}_{b}]}$	Acetate formation from AcCoA
$K_{8} = \frac{V_{max, glct} \cdot [{glc}_{b}]}{K_{m, glct} + [{glc}_{b}]}$	Gluconate transport
$K_{9} = D_{et}^{b} ([{et}_{e}] - [{et}_{b}])$	Diffusion ethanol (env $\leftrightarrow$ bact)
$K_{10} = D_{ace}^{y} ([{ace}_{y}] - [{ace}_{e}])$	Diffusion of acetate (bacteria $\to$ env)
Yeast
$K_{13} = D_{ace}^{y} ([{ace}_{e}] - [{ace}_{y}])$	Acetate diffusion
$K_{14} = \frac{V_{max, mlt}^{y} \cdot [{mlt}_{e}]}{K_{m, mlt}^{y} + [{mlt}_{e}]}$	Maltose transport
$K_{15} = \frac{V_{max, mlt}^{y} \cdot [{mlt}_{y}]}{K_{m, mlt}^{y} + [{mlt}_{y}]}$	Maltose metabolism $\to$ AcCoA
$K_{16} = \frac{V_{max, ace}^{y} \cdot [{ace}_{y}]}{K_{m, ace}^{y} + [{ace}_{y}]}$	Acetate $\to$ AcCoA
$K_{17} = \frac{V_{max, accoa}^{y} \cdot [{accoa}_{y}]}{K_{m, accoa}^{y} + [{accoa}_{y}]}$	AcCoA $\to$ Ethanol
$K_{18} = \frac{U_{m}^{y} \cdot [{accoa}_{y}]}{K_{s}^{y} + [{accoa}_{y}]}$	Growth
$K_{19} = D_{et}^{b} ([{et}_{y}] - [{et}_{e}])$	Diffusion of ethanol (yeast $\to$ env)
$Ω (p_{s}) = \sum_{i = 1}^{N} {(\hat{B} (t_{i}; p_{s}) - B_{i}^{data})}^{2}$	Scoring function, used to fit model simulation to experimental data

Using these reaction velocities, the ODEs for the consortium was written down, as displayed in the table below.

Ordinary differential equations of the model

Table 3: Ordinary differential equations used in the consortium model. where the positive sign denotes formation of the compoind and the negative sign denotes the consumption of the compound.

S.no	Equation	Description
Bacteria
1	$\frac{d [{glu}_{e}]}{d t} = - K_{1} X_{b}$	Bacteria glucose consumption rate
2	$\frac{d [{glu}_{b}]}{d t} = K_{1} - K_{2} - K_{4} - K_{3}$	Bacterial glucose balance
3	$\frac{d [cll]}{d t} = K_{2} X_{b}$	Cellulose formation rate
4	$\frac{d [{glc}_{b}]}{d t} = K_{3} - K_{8}$	Bacterial gluconate balance
5	$\frac{d [{accoa}_{b}]}{d t} = 1.4 K_{4} + K_{6} - K_{5} - K_{7}$	Acetyl-CoA balance in bacteria
6	$\frac{d [{et}_{b}]}{d t} = K_{9} - K_{6}$	Bacterial ethanol balance
7	$\frac{d [{et}_{e}]}{d t} = K_{19} X_{y} - K_{9} X_{b}$	Ethanol balance in environment
8	$\frac{d [{ace}_{b}]}{d t} = K_{7} - K_{20} - K_{sink} [{ace}_{b}]$	Acetate balance in bacteria. K_sink is a numerical value between 0 and 100
9	$\frac{d X_{b}}{d t} = K_{5} X_{b}$	Bacterial biomass growth
10	$\frac{d [{glc}_{e}]}{d t} = K_{8}$	Gluconate balance in environment
Yeast
11	$\frac{d [{mlt}_{e}]}{d t} = - K_{14} X_{y}$	Yeast maltose consumption
12	$\frac{d [{mlt}_{y}]}{d t} = K_{14} - K_{15}$	Yeast maltose balance
13	$\frac{d [{ace}_{y}]}{d t} = K_{13} - K_{16}$	Yeast acetate balance
14	$\frac{d [{accoa}_{y}]}{d t} = 1.4 K_{15} + K_{16} - K_{17} - K_{18}$	Yeast Acetyl-CoA balance
15	$\frac{d [{et}_{y}]}{d t} = K_{17} - K_{19}$	Yeast ethanol balance
16	$\frac{d [{ace}_{e}]}{d t} = K_{7} X_{b} - K_{13} X_{y}$	Acetate balance in environment
17	$\frac{d X_{y}}{d t} = K_{18} X_{y}$	Yeast biomass growth

Results

We first set up the model, through a code representing the growth of S. cerevisiae and K. sucrofermentans. We used the Michaelis-Menten parameters for the various rate limiting reactions, obtained from literature sources measuring individual reaction rates. The model was simulated at various initial glucose concentrations to identify any syntax or logical errors.

Our simulations indicated no logical flaws, we were able to conclude this as the initial output graphs indicated the uptake of glucose and other metabolites by the K. sucrofermentans and S. cerevisiae. However the Michaelis-Menten kinetic parameters needed to be adjusted to better match microbial growth curves. Based on biologically plausible ranges, the upper bounds of these parameters were set to 100 and lower bounds were set to 0.01. Using a looping function we ran the code iteratively 10,000 times with each iteration choosing random V_max and K_m values between the set bounds, to simulate the microbial growth.

We then used a scoring function to fit the model to laboratory experiments, of K. sucrofermentans growing on glucose, and S. cerevisiae growing on maltose. To better fit the model’s simulation to the data, we expanded the lower bounds, and used a Latin Hypercube Sampler (LHS) to ensure that parameters are chosen across the range evenly, thereby preventing clustering⁶. We filtered the best 100 simulations using this scoring function, to qualitatively evaluate the model’s simulation, based on which, we analysed the best fitting model (Figure 2).

**Figure 2:** Simulation of 100 best parameter sets of *K. sucrofermentans* (A) and *S. cerevisiae* (B) based on the scoring function, compared to experimental growth curve data.

From the simulation that best fit the growth data, we obtained the values for the Michaelis-Menten parameters. The bounds were then adjusted to a range of + and - 5% of the identified optimal value, and we repeated the previous step. We observed no major change to the output, and thus we conclude that the model is robust over minor changes to V_max and K_m values.

We established the cross-feeding mechanism to account for the uptake of acetate, production and subsequent diffusion of ethanol across the cell membrane by S. cerevisiae. Similarly we also established the uptake of ethanol, and production of and subsequent diffusion of acetate across the cell membrane by K. sucrofermentans. We then set the V_max and K_m values as constants from our previous data fitting, with the exception of parameters involved in the uptake and utilisation of acetate and ethanol by S. cerevisiae and K. sucrofermentans respectively. Due to the lack of data, we made an assumption that the diffusion of a compound across the cell membrane was dependant purely on the compound and independent of the host organism. That is, the efflux diffusion of ethanol from S. cerevisiae was identical to the influx or uptake of ethanol by K. sucrofermentans.

We then fit the simulation to the growth data for S. cerevisiae being grown on media with a composition of maltose and acetate, and K. sucrofermentans being grown on media with a composition of glucose and ethanol. We hypothesise that this would represent the consortium where S. cerevisiae produces ethanol and K. sucrofermentans produces acetate. By doing this, we were able to obtain the optimal V_max and K_m values for the cross-feeding mechanism. Using these we then simulated the consortium on differing ratios of inoculation of S. cerevisiae and K. sucrofermentans, to predict the growth of the co-culture.

Model derived best parameter values of K. sucrofermentans and S. cerevisiae

Table 4: Table reporting model derived best parameter values of K. sucrofermentans and S. cerevisiae after data fitting and optimisation.

S.No.	Parameter	Value	Description
Bacteria Parameters
1	$V_{max, glut}^{b}$	268.2451	Glucose transport
2	$K_{m, bglut}$	93.58049	Glucose transport constant
3	$V_{max, cll}$	93.40179	Cellulose formation
4	$K_{m, cll}$	9.22 ^-7	Cellulose formation constant
5	$V_{max, glc}$	2.182686	Gluconate formation
6	$K_{m, glc}$	72.83646	Gluconate formation constant
7	$V_{max, accoa}$	241.5771	Acetyl-CoA formation
8	$K_{m, accoa}$	0.940391	Acetyl-CoA formation constant
9	$μ^{b}$	2.991595	Growth
10	$K_{s, accoa}$	300.1778	Growth constant
11	$V_{max, aceb}$	183.4367	Acetate formation
12	$K_{m, aceb}$	42.76941	Acetate formation constant
13	$V_{max, glct}$	0.755053	Gluconate transport
14	$K_{m, glct}$	9.210217	Gluconate transport constant
15	$D_{et}^{b}$	480.87342537	Diffusion of ethanol into bacteria
16	$V_{max, et}$	23.75247306	Ethanol metabolism
17	$K_{m, et}$	2.94319963	Ethanol metabolism constant
Yeast Parameters
18	$V_{max, mltt}^{y}$	92.25304931	Maltose transport
19	$K_{m, mlt}^{y}$	59.72397984	Maltose transport constant
20	$V_{max, mlt}^{y}$	98.65120535	Maltose metabolism
21	$K_{m, mlt}^{y}$	3.78410763	Maltose metabolism constant
22	$V_{max, accoa}^{y}$	92.05153003	Acetyl-CoA metabolism
23	$K_{m, accoa}^{y}$	3.74583991	Acetyl-CoA metabolism constant
24	$μ^{y}$	1.58859315	Growth
25	$K_{s}^{y}$	89.2587033	Growth constant
26	$D_{ace}^{y}$	532.94794027	Diffusion of acetate into yeast
27	$V_{max, ace}^{y}$	8.92351645	Acetate metabolism
28	$K_{m, ace}^{y}$	359.25964698	Acetate metabolism constant

We observed an overproduction of acetate within the growth environment (Figure 3), after incorporating an acetate sink into our cross feeding mechanism, we were able to observe the growth of K. sucrofermentans and S. cerevisiae (Figure 4). We hypothesise, based on available literature, that through the cross-feeding mechanism, the yield of cellulose would increase. This is likely due to the shift on K. sucrofermentans’ dependancy on glucose for growth, which would theoretically reduce the need for glycolysis, allowing more glucose to be used for cellulose production. This hypothesis remains to be tested.

**Figure 3:** Model simulation of growth of *K. sucrofermentans* and *S. cerevisiae* in the consortium, indicating an overproduction of acetate, suggesting that potential modifications are required in the acetate consumption and utilisation by *S. cerevisiae*.

**Figure 4:** Model simulation of biomass growth in consortium with an acetate sink, indicating a qualitative match between experimental observation and the simulation.

Conclusions

From the initial model simulation, we noticed differences between the model predictions and laboratory data regarding the production of acetate and ethanol. The model predicts an overproduction of ethanol and acetate by the consortium, this affected the overall growth curves obtained by the model. We hypothesise that the predictions made by the model would improve when we incorporate an acetate sink.

Upon the incorporation of an acetate sink, the model predictions qualitatively matched what was observed in the laboratory, where the model predicted the successful growth of both microorganisms (Figure 4). We observed an additional inconsistency between the model and the wet-lab HPLC results, where we observed the production of ethanol, and consumption of acetate in the K. sucrofermentans monoculture. We hypothesise that this could be due to possible contamination within the sample. To test this hypothesis, we simulated the model, of K. sucrofermentans grown on glucose, with a low carry over concentration of maltose (0.5 g/L), and very low concentrations of S. cerevisiae representing the contamination. The simulation indicated the consumption of acetate by yeast, and the production of ethanol, which was in-line with what we observed in the lab.

(1)

Ryngajłło, M.; Jacek, P.; Cielecka, I.; Kalinowska, H.; Bielecki, S. Effect of Ethanol Supplementation on the Transcriptional Landscape of Bionanocellulose Producer Komagataeibacter Xylinus E25. Appl. Microbiol. Biotechnol. 2019, 103 (16), 6673–6688.

(2)

Jacek, P.; Silva, F. A. G. S. da; Dourado, F.; Bielecki, S.; Gama, M. Optimization and Characterization of Bacterial Nanocellulose Produced by Komagataeibacter Rhaeticus K3. Carbohydrate Polymer Technologies and Applications 2021, 2, 100022. https://doi.org/https://doi.org/10.1016/j.carpta.2020.100022.

(3)

Vanderwaeren, L.; Dok, veyda; Voordeckers, K.; Nuyts, S.; Verstrepen, K. J. Saccharomyces Cerevisiae as a Model System for Eukaryotic Cell Biology, from Cell Cycle Control to DNA Damage Response. Int. J. Mol. Sci. 2022, 23 (19), 11665.

(4)

Millard, P.; Enjalbert, B.; Uttenweiler-Joseph, S.; Portais, J.; Létisse, F. Control and Regulation of Acetate Overflow in Escherichia Coli. eLife 2021, 10, e63661. https://doi.org/10.7554/eLife.63661.

(5)

Olivares-Marin, I. K.; González-Hernández, J. C.; Regalado-Gonzalez, C.; Madrigal-Perez, L. A. Saccharomyces Cerevisiae Exponential Growth Kinetics in Batch Culture to Analyze Respiratory and Fermentative Metabolism. J. Vis. Exp. 2018, No. 139.

(6)

Boschini, M.; Gerosa, D.; Crespi, A.; Falcone, M. “LHS in LHS”: A New Expansion Strategy for Latin Hypercube Sampling in Simulation Design. SoftwareX 2025, 31, 102294. https://doi.org/https://doi.org/10.1016/j.softx.2025.102294.

Project Description

Project Design

Proof of Concept

Engineering

Implementation

Contribution

Wet Lab Overview

Functionalisation

Production Platform

Seed Coating & Application

Protocols

Part Collection

Dry Lab Overview

Ethanol Homeostasis Circuit Model

Consortium Model

Integrated Human Practices

Entrepreneurship

Members

Attributions

Sponsors

Notebook

Awards