ETHANOL HOMEOSTASIS CIRCUIT MODEL
Overview
Low concentrations of ethanol (1% (v/v)) have been shown to increase bacterial cellulose (BC) production1. To supply the ethanol for Komagataeibacter sucrofermentans, we decided to co-culture K. sucrofermentans with Saccharomyces cerevisiae. Natively, S. cerevisiae produces ethanol at a higher rate than K. sucrofermentans consumes ethanol1,2. Therefore, we developed an ethanol homeostasis circuit in S. cerevisiae that aims to maintain the external ethanol concentration at the optimal value of 1% (v/v) (Figure 1)1.
To guide the experimental implementation of the ethanol homeostasis circuit, a dynamic model based on ordinary differential equations (ODEs) was used. The model was constructed using parameter values from literature and was later fine-tuned to qualitatively describe experimental data of S. cerevisiae.
The model allowed us to assess whether our ethanol homeostasis circuit would indeed regulate the external ethanol concentration in the co-culture. Furthermore, it provided guidance on whether to implement the phosphoglycerate kinase 1 (PGK1) or phosphoglycerate mutase (GPM1) version of the ethanol homeostasis circuit in the wet lab. The model revealed important parameters that could be fine-tuned experimentally. We qualitatively assessed the effects of some of the important parameters on the behaviour of the circuit and investigated the effect of ethanol uptake by K. sucrofermentans and initial S. cerevisiae biomass on the external ethanol concentration. Overall, these findings provided us with a conceptual understanding of the ethanol homeostasis circuit and provided concrete directions to the successful implementation of the ethanol homeostasis circuit in the wet lab.
The ethanol homeostasis circuit
We designed our ethanol homeostasis circuit such that it contains a time-delayed negative feedback loop. A time-delayed negative feedback loop is a system in which A influences C through B and C influences A in the opposite direction as A influences C. In our circuit, ethanol (A) induces pTEV+ (C) through pTEV+ mRNA (B) and pTEV+ (C) inhibits (A) via PGK1 or GPM1 degradation (Figure 1). Mathematically, time-delayed negative feedback systems have been shown to reach steady states3. In other words, they should be able to keep the external ethanol concentration at a constant level. However, as the behaviour of an ethanol homeostasis circuit is sensitive, correct parameterisation is important4. To check if our circuit is indeed able to reach a steady state and to see which parameters are important, we built a dynamic model based on ODEs.
To create a time-delayed negative feedback, we used an ethanol-inducible promoter, inspired by the iGEM team of TU Münich from 2012 (Figure 1)5,6. When the external ethanol concentration is high, the ethanol concentration in S. cerevisiae is also high as a result of diffusion. At high ethanol concentrations, the promoter becomes active and expresses pTEV+. pTEV+ cuts an N-degron7,8. This N-degron is tagged to either PGK1 or GPM1. Both enzymes are involved in glucose fermentation. Once the N-degron is cleaved, the PGK1 or GPM1 enzyme is rapidly degraded7,8. We reasoned that this would lead to a reduction in the ethanol production rate by S. cerevisiae. So, when the external ethanol concentration is high, the ethanol production rate is reduced. The reverse was also hypothesised to be true.
The model
To be able to describe the ethanol homeostasis circuit, we constructed a dynamic model based on ODEs. The units of the model are mmol, L, and hours. We used the irreversible Michaelis-Menten (MM) rate law to describe how the state variables in the ethanol homeostasis circuit change over time. The irreversible MM rate law relates product formation to substrate concentration in the following way9:
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}).
For the reactions that are reversible, we used the reversible MM rate law10:
where P is the concentration of the product (mmol L^{-1}), S is the concentration of the substrate (mmol L^{-1}), K_{P} is the Michaelis constant of the product (mmol L^{-1}), K_{S} is the Michaelis constant of the substrate (mmol L^{-1}), V_{max} is the maximum reaction rate (mmol L^{-1} h^{-1}), and K_{eq} is the equilibrium constant.
To describe the diffusion of ethanol across the membrane of S. cerevisiae, we used the following rates, which assumes that both the external and internal ethanol concentration approach the homogeneous equilibrium concentration:
where v_{Eth_{int-diffusion}} and v_{Eth_{ext-diffusion}} are the respective ethanol fluxes (mmol L^{-1} h^{-1}), K_{diffusion} is the diffusion constant (h^{-1}), V_{ext} is the volume of the external environment (L), and V_{int} is the S. cerevisiae population volume (L).
To describe the population volume of S. cerevisiae over time, we used a Hill function. We let the growth depend on the ethanol flux, assuming that all ATP is produced during the 2-PG conversion step in our ethanol homeostasis circuit. The population volume of S. cerevisiae was modelled using the following equation:
Where V_{int} is the population volume of S. cerevisiae (L) and \mu is the growth rate (h^{-1}) defined as:
where \mu_{\mathrm{max}} is the maximum growth rate (h^{-1}), v_{Eth_{int}} is the flux of ethanol (mmol L^{-1} h^{-1}), and V_{{max,eth_{int}}} is the maximum ethanol flux (mmol L^{-1} h^{-1}).
We defined the transcription of pTEV+ to be dependent on the presence of ethanol, representing an ethanol-inducible promoter:
where \alpha_0 is the leaky transcription rate of pTEV+ (mmol L^{-1} h^{-1}), \alpha_1 is the transcription rate of pTEV+ when ethanol is maximised (mmol L^{-1} h^{-1}), Eth_{int} is the ethanol concentration inside S. cerevisiae (mmol L^{-1}), K_{d} is the dissociation constant of the promoter (mmol L^{-1}), and n is the cooperativity of the promoter.
To model the degradation of either PGK1 or GPM1 by pTEV+, we included the following term in the equations of PGK1 and GPM1:
where K_{d,TEVd} is the maximum degradation rate of pTEV+ (mmol L^{-1} h^{-1}), pTEV+ is the concentration of pTEV+ (mmol L^{-1}), and K_{TEV} is the saturation constant (mmol L^{-1}).
Importantly, the value of K_{d,TEVd} depends on the sequence of the N-degron that is fused to the enzyme. Therefore, this parameter can be adjusted during the experimental implementation of the ethanol homeostasis circuit. By changing the amino acid composition of the N-degron, the half-life of the tagged protein can vary from 2 minutes to 30 minutes11.
To let our model dynamically respond to degradation of either PGK1 or GPM1 by pTEV+, we explicitly specified the V_{max} for the DPG and G3P equations as [\mathrm{Enzyme}] \cdot K_{\mathrm{cat}}. We also included the term V_{\mathrm{int}}/V_{\mathrm{ext}} in the equation for the external glucose (Eq. 1) to correct for differences in volume between the external environment and the S. cerevisiae cells. The term -\mu \cdot [\text{metabolite}] was added to all S. cerevisiae related ODEs to correct for dilution due to population growth.
An overview of all ODE equations and their underlying rates can be found here.
Table 1: Ordinary differential equations of the model of the ethanol homeostasis circuit. The term V_{int}/V_{ext} corrects for the difference in volume between environments. pTEV+ mRNA expression was modelled as a saturable process. PGK1 and GPM1 degradation by pTEV+ was modelled as a saturable process. The term -\mu \cdot [\text{metabolite}] corrects for dilution due to S. cerevisiae population growth. Reaction rates are indicated by v followed by their appropriate annotation. vGlcint – DPG stands for the reaction rate of internal glucose to DPG. Reaction rates always start with a v. The “–” symbol in reaction rates should not be confused with the minus sign. Minus signs always appear in between two parameters and are separated by a space.
| Model equations | Eq. |
|---|---|
| \frac{d[Glc_{ext}]}{dt} = - \dfrac{V_{int}}{V_{ext}} \cdot v_{glc_{uptake}} | 1 |
| \frac{d[Glc_{int}]}{dt} = v_{glc_{uptake}} - v_{Glc_{int}-DPG} - \mu \cdot Glc_{int} | 2 |
| \frac{d[DPG]}{dt} = 2 \cdot v_{Glc_{int}- DPG} - v_{DPG-G3P} - \mu \cdot DPG | 3 |
| \frac{d[G3P]}{dt} = v_{DPG-G3P} - v_{G3P - 2\text{--}PG} - \mu \cdot G3P | 4 |
| \frac{d[2\text{-}PG]}{dt} = v_{G3P-2\text{--}PG} - v_{Eth_{int}} - \mu \cdot 2\text{--}PG | 5 |
| \frac{d[Eth_{int}]}{dt} = v_{Eth_{int}} + v_{Eth_{int-diffusion}} - \mu \cdot Eth_{int} | 6 |
| \frac{d[Eth_{ext}]}{dt} = v_{Eth_{ext-diffusion}} - k_{uptake} \cdot Eth_{ext} | 7 |
| \frac{d[\text{pTEV+}_{\text{mRNA}}]}{dt} = \alpha_0 + \alpha_1 \cdot \frac{{Eth_{int}}^n}{K_{d}^n + {Eth_{int}}^n} - k_{d,mRNA1} \cdot {pTEV+_{mRNA}} - \mu \cdot pTEV+_{mRNA} | 8 |
| \frac{d[\text{pTEV+}]}{dt} = \beta_1 \cdot \text{pTEV+}_{\text{mRNA}} - K_{d,TEV} \cdot \text{pTEV+} - \mu \cdot pTEV+ | 9 |
| \frac{d[\text{PGK1}_{\text{mRNA}}]}{dt} = \alpha_2 - k_{d,mRNA2} \cdot {PGK1_{mRNA}} - \mu \cdot PGK1_{mRNA} | 10 |
| \frac{d[\text{PGK1}]}{dt} = \beta_2 \cdot \text{PGK1}_{\text{mRNA}} - K_{d,PGK1} \cdot \text{PGK1} - K_{d,TEVd1} \cdot \frac{pTEV+}{K_{TEV} + pTEV+} \cdot PGK1 - \mu \cdot PGK1 | 11 |
| \frac{d[\text{GPM1}_{\text{mRNA}}]}{dt} = \alpha_3 - k_{d,mRNA3} \cdot {GPM1_{mRNA}} - \mu \cdot GPM1_{mRNA} | 12 |
| \frac{d[\text{GPM1}]}{dt} = \beta_3 \cdot \text{GPM1}_{\text{mRNA}} - K_{d,GPM1} \cdot \text{GPM1} - K_{d,TEVd2} \cdot \frac{pTEV+}{K_{TEV} + pTEV+} \cdot GPM1 - \mu \cdot GPM1 | 13 |
| \frac{dV_{int}}{dt} = \mu \cdot V_{int} | 14 |
| \frac{dV_{ext}}{dt} = - \mu \cdot V_{int} | 15 |
Table 2: Reaction rates of the ordinary differential equations used in the ethanol homeostasis circuit model. The V_{\text{max}} values of v_{\text{DPG--G3P}} and v_{\text{G3P--2-PG}} were explicitly modeled as [\text{PGK1}] \cdot k_{\text{cat,PGK1}} and [\text{GPM1}] \cdot k_{\text{cat,GPM1}}, respectively. The rates v_{\text{Eth}_{\text{int}}\text{-diffusion}} and v_{\text{Eth}_{\text{ext}}\text{-diffusion}} were modelled based on the principle of diffusion. The growth rate (\mu) was modelled to be directly proportional to v_{\text{Eth}_{\text{int}}}.
| Reaction rate | Eq. |
|---|---|
| v_{glc_{uptake}} = \frac{V_{max,glc_{int}}}{K_{Glc_{ext}}} \cdot \frac{1}{1 + \frac{Glc_{ext}}{K_{Glc_{ext}}} + \frac{Glc_{int}}{K_{Glc_{int}}}} \cdot \left( Glc_{ext} - \frac{Glc_{int}}{K_{eq, glc}} \right) | 16 |
| v_{Glc_{int}-DPG} = \frac{V_{\text{max,DPG}} \cdot Glc_{int}}{K_{m,glc_{int}} + Glc_{int}} | 17 |
| {v_{DPG-G3P}} = \frac{PGK1 \cdot k_{cat, PGK1}^+}{K_{DPG}} \cdot \frac{1}{1 + \frac{DPG}{K_{DPG}} + \frac{G3P}{K_{G3P, 1}}} \cdot \left(DPG - \frac{G3P}{K_{eq, PGK1}} \right) | 18 |
| {v_{G3P-2\text{--}PG}} = \frac{GPM1 \cdot k_{cat, GPM1}^+}{K_{G3P, 2}} \cdot \frac{1}{1 + \frac{G3P}{K_{G3P, 2}} + \frac{2\text{--}PG}{K_{2\text{--}PG}}} \cdot \left(G3P - \frac{2\text{--}PG}{K_{eq, GPM1}} \right) | 19 |
| v_{Eth_{int}} = \frac{V_{{max,eth_{int}}} \cdot {2\text{--}PG}}{K_{m,2\text{--}PG} + 2\text{--}PG} | 20 |
| v_{Eth_{int-diffusion}} = K_{diffusion} \cdot \frac{V_{ext} \cdot (Eth_{ext} - Eth_{int})}{V_{int} + V_{ext}} | 21 |
| v_{Eth_{ext-diffusion}} = K_{diffusion} \cdot \frac{V_{int} \cdot (Eth_{int} - Eth_{ext})}{V_{int} + V_{ext}} | 22 |
| \mu = \mu_{max} \cdot \frac{v_{Eth_{int}}}{V_{{max,eth_{int}}}} | 23 |
We based our parameter estimates and initial conditions on literature data from S. cerevisiae. K_{uptake} was initially set to 10^{-12} to simulate the ethanol homeostasis circuit in the absence of K. sucrofermentans. Moreover, both K_{d,TEVd1} and K_{d,TEVd2} were initially set to 10^{-12} in the null model to simulate native S. cerevisiae in the absence of the ethanol homeostasis circuit.
Table 3: Parameter estimates of the model of the ethanol homeostasis circuit. Due to facilitated diffusion, K_{\text{Glc,ext}} and K_{\text{Glc,int}} were given the same value, and K_{\text{eq,glc}} was set to one. When no literature data was available, K_m values were set to 10% of the reported equilibrium concentration of the respective substrate. \alpha_0 was set to 1/100 of \alpha_1. k_{\text{uptake}} was set to 10^{-12} to represent the absence of K. sucrofermentans.
| Parameter | Value | Unit | Interpretation | Ref. |
|---|---|---|---|---|
| V_{\text{max,Glc,int}} | 556 | mmol L^{-1} h^{-1} | Maximum S. cerevisiae glucose uptake rate | 12,13 |
| K_{\text{Glc,ext}} | 7.05 | mmol L^{-1} | Michaelis constant of external glucose | 14 |
| K_{\text{Glc,int}} | 7.05 | mmol L^{-1} | Michaelis constant of internal glucose (product) | 14 |
| K_{\text{eq,glc}} | 1 | – | Equilibrium constant of glucose uptake | 15 |
| V_{\text{max,DPG}} | 5,300 | mmol L^{-1} h^{-1} | Maximum DPG production rate | 15 |
| K_{m,\text{Glc,int}} | 0.27 | mmol L^{-1} | Michaelis constant of internal glucose (substrate) | 16 |
| k_{\text{cat,PGK1}}^+ | 21,240 | h^{-1} | Turnover number of PGK1 | 17 |
| K_{\text{DPG}} | 0.0044 | mmol L^{-1} | Michaelis constant of DPG | 17 |
| K_{\text{G3P,1}} | 0.76 | mmol L^{-1} | Michaelis constant of G3P (product) | 17 |
| K_{\text{eq,PGK1}} | 3,200 | – | Equilibrium constant of G3P production | 18 |
| k_{\text{cat,GPM1}}^+ | 31,800 | h^{-1} | Turnover number of GPM1 | 17 |
| K_{\text{G3P,2}} | 0.47 | mmol L^{-1} | Michaelis constant of G3P (substrate) | 17 |
| K_{\text{2-PG}} | 0.04 | mmol L^{-1} | Michaelis constant of 2-PG (product) | 17 |
| K_{\text{eq,GPM1}} | 0.19 | – | Equilibrium constant of 2-PG production | 18 |
| V_{\text{max,Eth,int}} | 5,300 | mmol L^{-1} h^{-1} | Maximum ethanol production rate | 15 |
| K_{m,\text{2-PG}} | 0.039 | mmol L^{-1} | Michaelis constant of 2-PG (substrate) | 19 |
| K_{diffusion} | 720 | h^{-1} | Diffusion rate of ethanol | 20 |
| K_{uptake} | 10^{-12} | h^{-1} | Ethanol uptake rate by K. sucrofermentans | |
| \alpha_0 | 3.76 x 10^{-9} | mmol L^{-1} h^{-1} | Leaky transcription rate of ethanol-inducible promoter\alpha_1 | 21–23 |
| \alpha_1 | 3.76x 10^{-7} | mmol L^{-1} h^{-1} | Transcription rate of ethanol-inducible promoter when ethanol is maximised | 21–23 |
| n | 2 | Cooperativity of ethanol-inducible promoter | ||
| K_d | 167.45 | mmol L^{-1} | Dissociation constant of ethanol-inducible promoter | |
| K_{d,\text{mRNA1}} | 2.47 | h^{-1} | Degradation rate of pTEV+ mRNA | 24 |
| \beta_1 | 74.64 | h^{-1} | Translation efficiency of pTEV+ mRNA | 25 |
| K_{d,TEV} | 6.91 | h^{-1} | Degradation rate of pTEV+ | 8 |
| \alpha_2 | 1.55 x 10^{-6} | mmol L^{-1} h^{-1} | Transcription rate of PGK1 mRNA | 21–23 |
| K_{d,\text{mRNA2}} | 2.45 | h^{-1} | Degradation rate of PGK1 mRNA | 24 |
| \beta_2 | 1853.44 | h^{-1} | Translation efficiency of PGK1 mRNA | 25 |
| K_{d,PGK1} | 0.042 | h^{-1} | Degradation rate of PGK1 | 26 |
| K_{d,TEVd1} | 10^{-12} | h^{-1} | Degradation rate of PGK1 by pTEV+ | 11 |
| K_{TEV} | 1.60 x 10^{-5} | mmol L^{-1} | Saturation constant of PGK1/GPM1 degradation by pTEV+ | 23,27 |
| \alpha_3 | 2.45 x 10^{-6} | mmol L^{-1} h^{-1} | Transcription rate of GPM1 mRNA | 21–23 |
| K_{d,\text{mRNA3}} | 2.45 | h^{-1} | Degradation rate of GPM1 mRNA | 24 |
| \beta_3 | 888.80 | h^{-1} | Translation efficiency of GPM1 mRNA | 25 |
| K_{d,GPM1} | 0.050 | h^{-1} | Degradation rate of GPM1 | 26 |
| K_{d,TEVd2} | 10^{-12} | h^{-1} | Degradation rate of GPM1 by pTEV+ | 11 |
| \mu_{\max} | 0.31 | h^{-1} | Maximum growth rate of S. cerevisiae | 28 |
Table 4: Initial conditions of the model of the ethanol homeostasis circuit. The initial external glucose concentration depended on the intended simulation. Values of 111, 277.5, and 555 mmol L^{-1} were used for 2%, 5%, 10% (w/v), respectively. Experiments assumed 0% (w/v) ethanol in the medium. The initial S. cerevisiae population volume in L was based on an OD of 0.1. PGK1 and GPM1 were assumed to start at their equilibrium concentrations, ignoring the effects of pTEV+ degradation and dilution.
| State variable | Initial value | Unit | Comment | Ref |
|---|---|---|---|---|
| External glucose | 111, 277.5, 555 | mmol L^{-1} | 2%, 5%, and 10% (w/v) glucose | |
| Internal glucose | 2.7 | mmol L^{-1} | 16 | |
| DPG | 0.03 | mmol L^{-1} | 19 | |
| G3P | 0.39 | mmol L^{-1} | 19 | |
| 2-PG | 0.39 | mmol L^{-1} | 19 | |
| Internal ethanol | 0 | mmol L^{-1} | ||
| External ethanol | 0 | mmol L^{-1} | 0% (w/v) ethanol medium | |
| pTEV+ mRNA | 0 | mmol L^{-1} | Ethanol induced | |
| pTEV+ | 0 | mmol L^{-1} | ||
| PGK1 mRNA | \frac{\alpha2}{K_{d,mRNA2}} | mmol L^{-1} | Start at equilibrium | |
| PGK1 | \frac{\beta2 \cdot \frac{\alpha2}{K_{d,mRNA2}}}{K_{d,PGK1}} | mmol L^{-1} | Start at equilibrium | |
| GPM1 mRNA | \frac{\alpha3}{K_{d,mRNA3}} | mmol L^{-1} | Start at equilibrium | |
| GPM1 | \frac{\beta3 \cdot \frac{\alpha3}{K_{d,mRNA3}}}{K_{d,GPM1}} | mmol L^{-1} | Start at equilibrium | |
| S. cerevisiae population volume | 6.536 x 10^{-8} | L | OD of 0.1 | 23,29 |
| External volume | 200 x 10^{-6} | L | 200 \muL assay |
Results
We first simulated the model using three different initial glucose concentrations to see if all the state variables behave as expected. Importantly, we initially used negligible values for K_{d,TEVd1} and K_{d,TEVd2} to simulate native S. cerevisiae without the ethanol homeostasis circuit implemented. We see that the state variables behave as expected (Figure 2). The external glucose concentration was predicted to be depleted after approximately 120 to 180 hours, depending on the initial glucose concentration. The internal glucose was immediately converted into DPG, which was converted into G3P. Both DPG and G3P seem to accumulate at first. G3P was converted into 2-PG, which was quickly converted into internal ethanol. The internal ethanol quickly diffused into the external environment. For the 2% and 5% (w/v) conditions, the ethanol concentration stopped increasing after approximately 150 to 180 hours. At this time, all glucose had been converted into ethanol. As expected, the TEV and TEV mRNA increased over time, until they approached their equilibrium. PGK1, PGK1 mRNA, GPM1, and GPM1 mRNA initially showed a slight decrease in concentration due to population growth of S. cerevisiae. When the population growth stopped, the concentrations approached the equilibrium again.
The model seemed to be biologically plausible. However, we noted that the timescale of the model was quite slow (> 200 hours) compared to real biological experiments. To confirm this hypothesis, we collected experimental S. cerevisiae glucose consumption, ethanol production, and growth data for a timespan of 24 hours. A comparison of our initial model simulation with the experimental data indeed confirmed that our initial model simulation was too slow (Figure 2 & 3). To compare our model with the experimental data, the S. cerevisiae population volume was rescaled from L to Optical Density (OD). Despite the poor fit when compared at the same timescale, we noted strong similarities between the model simulated for 240 hours and the experimental data (Figure 2 & 3). The experimental data seemed to function seven times faster than our model. Consequently, we rescaled the time in our model such that rates were seven times quicker. Moreover, we decreased the V_{\text{max,Eth,int}} parameter from 5,300 to 3,500 to quantitatively improve the fit. With these changes, the model could qualitatively describe the experimental data (Figure 3). Nevertheless, the S. cerevisiae population volume overshot. The latter may be due to the absence of a carrying capacity in the ODEs. Due to time constraints, we did not further investigate this.
To check if our ethanol homeostasis circuit indeed reduces the ethanol production rate of S. cerevisiae compared to the null model, we changed either K_{d,TEVd1} or K_{d,TEVd2} to simulate the PGK1 and GPM1 model, respectively. As we were also interested in the exact parameter value that provides us with the desired ethanol production behaviour, we specified a range for K_{d,TEVd1} and K_{d,TEVd2} and performed 10,000 optimisations using Latin Hypercube Sampling. We also specified ranges for other experimentally adjustable parameters related to the ethanol-inducible promoter and pTEV+ expression. A mean square error-based function was used as scoring function, where the mean of an artificial dataset was set to the desired ethanol concentration of 171 mmol L^{-1}, corresponding to 1% (v/v). This way, we forced the model to approach the desired ethanol concentration. The standard error of the artificial data was set to 1 for all timepoints. The optimisations showed that the ethanol homeostasis circuit indeed reduces the ethanol production rate by reducing either the PGK1 or GPM1 concentration, depending on the version of the ethanol homeostasis circuit (Figure 4). However, the model did not predict any difference between the PGK1 and GPM1 version of the ethanol homeostasis circuit in terms of external ethanol concentration. Consequently, we decided to focus only on the implementation of the PGK1 version in the wet lab.
Table 5: Bounds used for the PGK1 and GPM1 optimisation.
| Parameter | Lower Bound | Upper Bound | Unit | Comment | Ref. |
|---|---|---|---|---|---|
| \alpha_0 | 5.82 x 10^{-10} | 3.76 x 10^{-9} | mmol L^{-1} h^{-1} | 100th of \alpha_1 | 21–23 |
| \alpha_1 | 5.82 x 10^{-8} | 3.76 x 10^{-7} | mmol L^{-1} h^{-1} | 10th to 90th percentile transcriptional activities S. cerevisiae | 21–23 |
| n | 1 | 5 | |||
| K_d | 21.4 | 314 | mmol L^{-1} | Lower bound is 10th of desired ethanol concentration | |
| K_{d,\text{mRNA1}} | 1.66 | 2.97 | h^{-1} | 10th to 90th percentile mRNA decay S. cerevisiae | 24 |
| \beta_1 | 35.57 | 946.15 | h^{-1} | 10th to 90th percentile translation efficiencies S. cerevisiae | 25 |
| K_{d,TEVd1} | 1.39 | 20.79 | h^{-1} | 11 | |
| K_{TEV} | 1.60 x 10^{-6} | 2.78 x 10^{-5} | mmol L^{-1} | 10th of 10th and 90th percentile protein abundance in S. cerevisiae | 23,27 |
| K_{d,TEVd2} | 1.39 | 20.79 | h^{-1} | 11 |
Where \mathrm{\Omega} is the score, \theta is the parameter set, \hat{y} is the artificial data, y is the model simulation, \hat{\sigma} is the standard error of the artificial data, t_i is the time (h) at index i, and n is the last index of the simulated time.
Table 6: Optimisation results for the best PGK1 and GPM1 model.
| Parameter | PGK1 | GPM1 | Unit | Ref. |
|---|---|---|---|---|
| α0 | 5.82 × 10−10 | 5.82 × 10−10 | mmol L−1 h−1 | 21–23 |
| α1 | 3.76 × 10−7 | 3.76 × 10−7 | mmol L−1 h−1 | 21–23 |
| n | 4.61 | 4.76 | ||
| Kd | 180.25 | 165.95 | mmol L−1 | |
| Kd, mRNA1 | 2.57 | 2.97 | h−1 | 24 |
| β1 | 56.42 | 35.57 | h−1 | 25 |
| Kd, TEVd1 | 10.23 | 1 × 10−12 | h−1 | 11 |
| KTEV | 2.24 × 10−5 | 2.78 × 10−5 | mmol L−1 | 23,27 |
| Kd, TEVd2 | 1 × 10−12 | 20.79 | h−1 | 11 |
To see which parameters are important to fine-tune during the experimental implementation, we visualised the distribution of parameter estimates of the 1% best-scoring optimisations for each parameter (Figure 5). If the parameter does not influence the desired behaviour, a uniform distribution is expected. Conversely, if the parameter does influence the desired behaviour, a non-uniform distribution is expected. In our case, K_{tev} and K_{d,\text{mRNA1}} do not influence the desired behaviour (Figure 5). The leaky transcription rate of the promoter (\alpha_0), the maximum transcription rate of the promoter (\alpha_1), the cooperativity of the promoter (n), the dissociation constant of the promoter (K_d), the translation rate of pTEV+, and the N-degron degradation rate (K_{d,TEVd1}) all seem to influence the behaviour of the ethanol homeostasis circuit. As many of these parameters are related to the ethanol-inducible promoter, we are planning to perform ethanol-inducible promoter dynamic range assays in the lab to characterise these parameters of several promoters.
To better understand the ethanol homeostasis circuit, we decided to visualise the qualitative effect of the dissociation constant (K_d). We chose the focus on the dissociation constant, as this parameter is tuneable by modifying the sequence of the promoter30. As expected, low dissociation constants result in early inhibition of the ethanol production rate, whereas high dissociation constants result in late inhibition (Figure 6). These results confirm that the dissociation constant is an important parameter. These results also show that the ethanol homeostasis circuit does not stop the ethanol production by S. cerevisiae completely. After PGK1 inhibition by pTEV+ there is still a small amount of ethanol production.
So far, we did not include ethanol uptake by K. sucrofermentans (K_{uptake}) in our model. However, in the consortium K. sucrofermentans is present and will influence the external ethanol concentration. Therefore, we decided to model the effect of different ethanol uptake rates by K. sucrofermentans. We were also interested in whether the uptake of ethanol by K. sucrofermentans could counteract the small amount of leaky ethanol production identified above thereby helping to stabilise the external ethanol concentration. The simulations show that the ethanol uptake rate by K. sucrofermentans indeed influences the external ethanol concentration (Figure 7). Moreover, specific ethanol uptake rates help stabilise the external ethanol concentration at 171 mmol L^{-1}, corresponding to 1% (v/v). When the uptake becomes too large, the external ethanol concentration does not reach the desired concentration at all. These findings highlight the importance of including K. sucrofermentans in subsequent models that aim to provide quantitative recommendations for the experimentally tuneable parameters.
In the simulation above, we set a fixed value for K. sucrofermentans ethanol uptake rate that only depended on the external ethanol concentration. However, we suspected that the external ethanol uptake rate of K. sucrofermentans does not only depend on the external ethanol concentration but also on the population volume of K. sucrofermentans. As we did not include the population volume of K. sucrofermentans in our model, we could not directly test this hypothesis. However, we could test the effect of different initial S. cerevisiae population volumes on the reaction rates. We found that initial population volume was positively correlated with the reaction rates (Figure 8). This suggests that the ethanol uptake by K. sucrofermentans should be dependent on the population volume. To do so, the population volume of K. sucrofermentans must be explicitly modelled in subsequent models.
Conclusion
The model was able to qualitatively describe native S. cerevisiae data. We used the model to simulate the ethanol homeostasis circuit we developed. Model simulations suggest that the ethanol homeostasis circuit indeed reduces the ethanol production rate by S. cerevisiae. Moreover, our results suggest that there is no difference between the PGK1 or GPM1 version of the ethanol homeostasis circuit. Based on this, we decided to only continue with the PGK1 version in the wet lab. We used the model to identify important parameters for controlling the external ethanol concentration at 1% (v/v). We found that many important parameters were related to the ethanol-inducible promoter. Therefore, we decided to plan promoter dynamic range assays in the wet lab for several ethanol-inducible promoters.
To improve our understanding of the ethanol homeostasis circuit, we visualised the effect of the dissociation constant (one of the important parameters) on the behaviour of the ethanol homeostasis circuit. Once we understood the ethanol homeostasis circuit, we included ethanol uptake by K. sucrofermentans in our model. We showed that the inclusion of the ethanol uptake by K. sucrofermentans impacts the external ethanol concentration and that the uptake may help stabilise the external ethanol concentration. Lastly, we showed that the population volumes of both S. cerevisiae and K. sucrofermentans are important to be included in a future model. Overall, the model guided the experimental implementation of the ethanol homeostasis circuit in S. cerevisiae, yet the model predictions remain to be tested. To understand how the ethanol homeostasis circuit functions in the co-culture, future research could implement this model within the consortium model.