# Computational Modelling of Metabolic Burden and Substrate Toxicity in Escherichia coli Carrying a Synthetic Metabolic Pathway

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Laboratory Methods

#### 2.1.1. Bacterial Strains and Plasmids

#### 2.1.2. Preparation of Pre-Induced Cells

#### 2.1.3. Short Term Toxicity Test

#### 2.1.4. Growth Test

#### 2.1.5. Glycerol Analysis

#### 2.2. Parameter Constraints

#### 2.3. Biochemical Model

#### 2.4. Inverse Problem

#### 2.4.1. Parameter Estimation and Regression

#### 2.4.2. Parameter Synthesis and Robustness Monitoring

#### 2.5. Analysis Workflow

#### 2.5.1. General Assumptions

- We assume the total inducer concentration to be constant in the time frame of our interest. An inducer is supposed to have a function of an input parameter, and it would be an inadequate parameter should it be adjusted spontaneously over time. Otherwise, the inducer degradation rate would be needed either found in literature or extracted from experimental data.
- The workflow is limited to protease-deficient bacterial strains (e.g., E. coli BL21). In particular, we assume the total concentration of every enzyme affecting the studied pathway is constant in the time frame of our interest. Moreover, no influx of the enzymes is permitted as a consequence of time-limited induction phase where the proteosynthesis takes place [50,51]. Additional synthetic processes are considered negligible in a microbial population stressed enough by the massive expression of (heterologous) genes during induction.
- There occurs a metabolic burden effect caused by the heterologous genes expression during the induction process and possibly by the presence of an inducer itself which in a combined way affects the bacterial growth rate.
- Finally, we assume the bacterial population is in the stationary phase after the induction process is finished.

#### 2.5.2. Workflow Description

#### 2.6. Software Tools

## 3. Results and Discussion

#### 3.1. Extended Assumptions

- We define a new variable called Bact standing for CDW (g/L) of E. coli population taken as 0.39 g/L = 1 ${\mathrm{OD}}_{600}$ [24].
- We assume IPTG to be the only inducer for the synthetic pathway. Concentration of IPTG is considered to be constant in the given time frame.
- Reversible reactions in the TCP-degradation pathway are considered negligible.
- The initial concentration of substances (e.g., ${\mathrm{TCP}}_{0}$, ${\mathrm{GLY}}_{0}$, ${\mathrm{IPTG}}_{0}$) and the population (i.e., ${\mathrm{Bact}}_{0}$) determines the input for the system.
- Dynamics of individual enzymes is approximated as a constant function of time in the given time frame. Moreover, enzymes dynamics is considered to be independent on the size of the bacterial population in the given time frame.
- Total conversion of TCP into GLY is assumed to occur in a sufficiently long time reflecting the known behaviour of the pathway.
- Viability of the bacterial population is given as the function of the pathway compounds toxicity, metabolic burden and the presence of nutrients (i.e., GLY).
- Toxic effects of the pathway compounds are considered to be mutually independent.
- Glycerol is the only assumed nutrient.
- We assume natural degradation (death rate) of the bacterial population.

#### 3.2. Workflow Input: Synthetic TCP Degradation Pathway

#### 3.3. Step 1: Enzymatic Space Settings and Reduction

#### 3.4. Step 2: Integration with Population Growth

#### 3.5. Step 3: Model Dynamics Exploration

**Concentration of IPTG in mM**: IPTG is an obvious candidate for tuning because many aspects of the model depend on it and it can be controlled easily in the experimental environment (since its concentration is considered constant during the experiment, it can be referred by its initial concentration, denoted ${\mathbf{IPTG}}_{0}$).**Size of bacterial population (Bact) in g/L**: The initial population size, denoted ${\mathbf{Bact}}_{0}$, makes the crucial input of the model and it affects the model output—the final population size (reached in a given time). In general, the initial population size can be controlled in experiments.**Initial concentration of TCP (**${\mathbf{TCP}}_{0}$**) in mM**: The key input of the model that must be set in order to make the modelled metabolic pathway work; it can be easily set to any arbitrary value during the experiments.**Death rate of the population (**${\mathit{\gamma}}_{\mathbf{Bact}}$**) in**${\mathbf{h}}^{-1}$: The death rate is considered as a parameter because we are interested in the dynamics of the microbial culture and the effects affecting the growth.

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## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

IPTG | isopropyl-beta-D-thiogalactopyranoside |

TCP | 1,2,3-trichloropropane |

DCP | 2,3-dichloropropane-1-ol |

ECH | epichlorohydrin |

CPD | 3-chloropropane-1,2-diol |

GDL | glycidol |

GLY | glycerol |

DhaA | haloalkane dehalogenase |

EchA | epoxide hydrolase |

HheC | haloalcohol dehalogenase |

MM | Michaelis-Menten |

CDW | cell dry weight |

ODE | ordinary differential equations |

MCMC | Markov chain Monte Carlo |

GUI | graphical user interface |

CLI | command-line interface |

## Appendix A. The Experimental Data Used for the Fitting of Functions Explaining the Enzymes Concentration Determined at Respective IPTG Concentrations

IPTG (mM) | Content in Cell-Free Extract (%) * | DhaA31 (Relative Portion) | HheC (Relative Portion) | EchA (Relative Portion) | Total Mass ($\frac{\mathbf{mg}}{10\phantom{\rule{3.33333pt}{0ex}}\mathbf{mL}}$) |
---|---|---|---|---|---|

1.0 | 50 | 0.19 | 0.39 | 0.42 | 4.188 |

0.2 | 48 | 0.17 | 0.40 | 0.43 | 4.02 |

0.05 | 40 | 0.13 | 0.38 | 0.49 | 3.350 |

0.025 | 40 | 0.12 | 0.41 | 0.47 | 3.350 |

0.01 | 22 | 0.16 | 0.36 | 0.48 | 1.843 |

0.0 | 15 | 0.20 | 0.33 | 0.47 | 1.256 |

IPTG (mM) | Content in Cell-Free Extract (%) * | DhaA31 (Relative Portion) | HheC (Relative Portion) | EchA (Relative Portion) | Total Mass ($\frac{\mathbf{mg}}{10\phantom{\rule{3.33333pt}{0ex}}\mathbf{mL}}$) |
---|---|---|---|---|---|

1.0 | 60 | 0.15 | 0.39 | 0.46 | 3.890 |

0.2 | 62 | 0.15 | 0.39 | 0.46 | 4.020 |

0.05 | 50 | 0.13 | 0.38 | 0.49 | 3.242 |

0.025 | 50 | 0.11 | 0.37 | 0.52 | 3.242 |

0.01 | 42 | 0.11 | 0.37 | 0.52 | 2.723 |

0.0 | 15 | 0.17 | 0.40 | 0.44 | 0.973 |

**Figure A1.**Sodium dodecyl sulfate polyacrylamide gel electrophoresis of cell-free extracts obtained from E. coli deg31 cells induced with various concentrations of IPTG ((

**M**) protein marker (116, 66.2, 45, 35, 25, 18.4, and 14.4 kDa)): (

**a**) 0.0 mM IPTG; (

**b**) 0.01 mM IPTG; (

**c**) 0.025 mM IPTG; (

**d**) 0.05 mM IPTG; (

**e**) 0.2 mM IPTG; and (

**f**) 1.0 mM IPTG. Bands of DhaA31 (34 kDa), HheC (29 kDa) and EchA (35 kDa) are marked.

IPTG | Total Mass | DhaA31 | HheC | EchA | DhaA31 | HheC | EchA |
---|---|---|---|---|---|---|---|

(mM) | $\left(\frac{\mathbf{mg}}{10\phantom{\rule{3.33333pt}{0ex}}\mathbf{mL}}\right)$ | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | (mM) ^{#} | (mM) ^{#} | (mM) ^{#} |

1.0 | 4.188 | 79.6 | 163.3 | 175.9 | 2.24 × 10^{−3} | 5.57 × 10^{−3} | 4.82 × 10^{−3} |

0.2 | 4.02 | 68.3 | 160.8 | 172.9 | 1.92 × 10^{−3} | 5.48 × 10^{−3} | 4.74 × 10^{−3} |

0.05 | 3.350 | 43.6 | 127.3 | 164.2 | 1.22 × 10^{−3} | 4.34 × 10^{−3} | 4.50 × 10^{−3} |

0.025 | 3.350 | 40.2 | 137.4 | 157.5 | 1.13 × 10^{−3} | 4.68 × 10^{−3} | 4.32 × 10^{−3} |

0.01 | 1.843 | 29.5 | 66.3 | 88.4 | 8.29 × 10^{−4} | 2.26 × 10^{−3} | 2.43 × 10^{−3} |

0.0 | 1.256 | 25.1 | 41.5 | 59.0 | 7.06 × 10^{−4} | 1.41 × 10^{−3} | 1.62 × 10^{−3} |

^{#}Calculated as $\left({\displaystyle \frac{\mathrm{mg}}{\mathrm{Da}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{L}}}\right)$.

IPTG | Total Mass | DhaA31 | HheC | EchA | DhaA31 | HheC | EchA |
---|---|---|---|---|---|---|---|

(mM) | $\left(\frac{\mathbf{mg}}{10\phantom{\rule{3.33333pt}{0ex}}\mathbf{mL}}\right)$ | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | $\left(\frac{\mathbf{mg}}{\mathbf{L}}\right)$ * | (mM) ^{#} | (mM) ^{#} | (mM) ^{#} |

1.0 | 3.89 | 58.4 | 151.7 | 179.0 | 1.64 × 10^{−3} | 5.17 × 10^{−3} | 4.91 × 10^{−3} |

0.2 | 4.02 | 60.3 | 156.8 | 184.9 | 1.69 × 10^{−3} | 5.34 × 10^{−3} | 5.07 × 10^{−3} |

0.05 | 3.242 | 42.1 | 123.2 | 158.9 | 1.18 × 10^{−3} | 4.20 × 10^{−3} | 4.36 × 10^{−3} |

0.025 | 3.242 | 35.7 | 120.0 | 168.6 | 1.00 × 10^{−3} | 4.09 × 10^{−3} | 4.62 × 10^{−3} |

0.01 | 2.723 | 30.0 | 100.8 | 141.6 | 8.42 × 10^{−4} | 3.44 × 10^{−3} | 3.88 × 10^{−3} |

0.0 | 0.973 × 10^{−1} | 16.5 | 38.9 | 42.8 | 4.65 × 10^{−4} | 1.33 × 10^{−3} | 1.17 × 10^{−3} |

^{#}Calculated as $\left({\displaystyle \frac{\mathrm{mg}}{\mathrm{Da}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{L}}}\right)$.

DhaA31 (Da) | HheC (Da) | EchA (Da) |
---|---|---|

35,576.37 | 29,333.07 | 36,465.11 |

IPTG | DhaA31 (mM) | HheC (mM) | EchA (mM) | |||
---|---|---|---|---|---|---|

(mM) | Median | stdev | Median | stdev | Median | stdev |

1.0 | 1.94 × 10^{−3} | 4.22 × 10^{−4} | 5.37 × 10^{−3} | 2.79 × 10^{−4} | 4.87 × 10^{−3} | 5.97 × 10^{−5} |

0.2 | 1.81 × 10^{−3} | 1.60 × 10^{−4} | 5.41 × 10^{−3} | 9.69 × 10^{−5} | 4.91 × 10^{−3} | 2.34 × 10^{−4} |

0.05 | 1.20 × 10^{−3} | 2.78 × 10^{−5} | 4.27 × 10^{−3} | 9.90 × 10^{−5} | 4.43 × 10^{−3} | 1.03 × 10^{−4} |

0.025 | 1.07 × 10^{−3} | 9.02 × 10^{−5} | 4.39 × 10^{−3} | 4.19 × 10^{−4} | 4.47 × 10^{−3} | 2.16 × 10^{−4} |

0.01 | 8.35 × 10^{−4} | 9.45 × 10^{−6} | 2.85 × 10^{−3} | 8.30 × 10^{−4} | 3.15 × 10^{−3} | 1.03 × 10^{−3} |

0.0 | 5.85 × 10^{−4} | 1.71 × 10^{−4} | 1.37 × 10^{−3} | 6.15 × 10^{−5} | 1.40 × 10^{−3} | 3.15 × 10^{−4} |

## Appendix B. The Origin of the Death Rate Coefficient

**Table A7.**The death rate ($\gamma $) coefficients evaluated from different growth rates (g) using the generation time (${\mathrm{t}}_{\mathrm{g}}$) and percentage death rate per generation (% p.g.).

g (${\mathbf{h}}^{-1}$) | ${\mathbf{t}}_{\mathbf{g}}$ (h) | $\mathit{\gamma}$ (${\mathbf{h}}^{-1}$) as 0.5% p.g. | $\mathit{\gamma}$ (${\mathbf{h}}^{-1}$) as 1% p.g. |
---|---|---|---|

0.26 | 2.58 | $1.938\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | $\mathbf{3.876}\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{\mathbf{10}}^{-\mathbf{3}}$ |

0.07 | 9.29 | $\mathbf{5.382}\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{\mathbf{10}}^{-\mathbf{4}}$ | $1.076\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ |

## Appendix C. The List of Considered Inhibition Models

Monod + Hill [27]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k{\left[X\right]}^{n}}{{{K}_{\mathrm{i}}}^{n}+{\left[X\right]}^{n}}$ | (A3) |

Monod + Aiba-Edward [65]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k\left[X\right]}{{K}_{\mathrm{X}}+\left[X\right]}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}exp\left(\frac{-\left[X\right]}{{K}_{\mathrm{i}}}\right)$ | (A4) |

Monod + Andrews [66]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k}{(1+\frac{{K}_{X}}{\left[X\right]})(1+\frac{\left[X\right]}{{K}_{\mathrm{i}}})}$ | (A5) |

Monod + Haldane-Andrews [66]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k\left[X\right]}{{K}_{X}+\left[X\right]+\frac{{\left[X\right]}^{2}}{{K}_{\mathrm{i}}}}$ | (A6) |

Monod + Monod: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k\left[X\right]}{{K}_{X}+\left[X\right]}$ | (A7) |

Monod + Moser [64]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k{\left[X\right]}^{n}}{{K}_{X}+{\left[X\right]}^{n}}$ | (A8) |

Monod + Tessier [63]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}k\left(\right)open="("\; close=")">1-exp\left(\frac{-\left[X\right]}{{K}_{\mathrm{i}}}\right)$ | (A9) |

Monod + Tessier-type [65]: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}-\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}k\left(\right)open="("\; close=")">exp\left(\frac{-\left[X\right]}{{K}_{\mathrm{i}}}\right)-exp\left(\frac{-\left[X\right]}{{K}_{X}}\right)$ | (A10) |

competitive inhibition: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}(1+\frac{\left[X\right]}{{K}_{\mathrm{i}}})+\left[S\right]}$ | (A11) |

non-competitive inhibition: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}}{1+\frac{\left[X\right]}{{K}_{\mathrm{i}}}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{\left[S\right]}{{K}_{S}+\left[S\right]}$ | (A12) |

uncompetitive inhibition: | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right](1+\frac{\left[X\right]}{{K}_{\mathrm{i}}})}$ | (A13) |

non-competitive inhibition using negative Hill | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{{K}_{\mathrm{i}}}^{n}}{{{K}_{\mathrm{i}}}^{n}+{\left[X\right]}^{n}}$ | (A14) |

non-competitive inhibition using negative Moser | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{K}_{\mathrm{i}}}{{K}_{\mathrm{i}}+{\left[X\right]}^{n}}$ | (A15) |

non-competitive exponential inhibition | $\frac{d\left[B\right]}{dt}=\left[B\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{\mu}_{\mathrm{max}}\left[S\right]}{{K}_{S}+\left[S\right]}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{k}{{{K}_{\mathrm{i}}}^{\left[X\right]}}$ | (A16) |

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**Figure 1.**Model of metabolic pathway for biodegradation of TCP. A general scheme of an enzymatic metabolic pathway for biodegradation of TCP into GLY. Note that DhaA31 produces two different enantiomers of 2,3-dichloropropane-1-ol (DCP) with similar rate. However, enzyme HheC has notably different enantioselectivity with them. It is also worth noting that enzymes HheC and EchA are employed twice in the pathway. Other intermediates are epichlorohydrin (ECH), 3-chloropropane-1,2-diol (CPD), and glycidol (GDL).

**Figure 2.**Definition of the ramp function. (

**Right**) A mathematical definition of an increasing ramp function as used in our workflow. Parameters a and b are usually set to values 0 and 1, respectively; and vice versa for decreasing version. Values ${t}_{1}$ and ${t}_{2}$ typically represent some significant thresholds on x. (

**Left**) Graphical description of the same function.

**Figure 3.**Definition of the Heaviside step function. (

**Right**) A mathematical definition of an increasing Heaviside step function as used in our workflow. Parameters a and b are usually set to values 0 and 1, respectively; and vice versa for decreasing version. Value t typically represents an important threshold in the domain of x. (

**Left**) Graphical description of the function—specifically, the increasing version.

**Figure 4.**Proposed model scheme. A schematic description of the novel ODE model with highlighted extending partitions. Each partition (i.e., module) represents a linearly independent part of a particular equation. Each module has a unique semantic function. Note that some of the modules are used in more than one equation. The coloured modules represent entirely new parts of the final model and the grey modules were extended in this study. This modular feature was necessary to handle the fitting of such a complex model to the experimental data.

**Figure 5.**Results of fitting to enzymes concentration data. Three plots show individual results of fitting to concentration data measured in various starting concentration levels of the inducer (${\mathrm{IPTG}}_{0}$)—0.0, 0.01, 0.025, 0.05, 0.2 and 1.0 (mM)—for three different enzymes: DhaA (

**top left**); HheC (

**top right**); and EchA (

**bottom**). The experimental data are pictured as points and the results—fitted curves—are pictured as lines. Both axes show a concentration level in mM, the inducer on the x-axis and the particular enzyme on the y-axis of the particular plot. Error bars represent standard deviation values calculated from two independent experiments.

**Figure 6.**Results of fitting to population growth data. Two plots showing different results of fitting to population growth data from two points of view: (

**left**) the particular results for CDW starting at 0.0429 and ending at 0.468 g/L after 10 h of growth; and (

**right**) the result for the substrate utilisation only starting at 10.12 and ending at 0.07 mM after 10 h of growth. The experimental data are pictured as points with standard error bars, the dashed lines show simulation data for initial values of Monod function (i.e., initial point of fitting), the solid lines show the results of fitting (Section 2.4.1) and the dotted lines represent the final results optimised by MCMC method of the FME package (Section 2.4.1), which show the best agreement with the experimental data. The x-axes show time of experiment in hours; the y-axis of the right plot shows the concentration of GLY in mM; and the y-axis of the left plot shows CDW in g/L of bacterial population (Bact).

**Figure 7.**Bacterial population growth data reflecting concentration of inducer. The plot shows curves of population growth on GLY for cultures reflecting different concentrations of IPTG prepared according to Section 2.1.4. All cultures—carrying plasmids with heterologous metabolic pathway—started at the same value but ended with different size of population depending on the initial concentration of the inducer (${\mathrm{IPTG}}_{0}$): 0, 0.01, 0.05, 0.2, and 1 (mM). Note that the rising amount of IPTG led to progressive inhibition of bacterial growth. The most interesting is the big step from 0.01 to 0.05 of IPTG. This notable difference in the population on the relatively small interval of IPTG values and minimal changes in the population for the rest of IPTG concentrations shows the high sensitivity of the population to ${\mathrm{IPTG}}_{0}$.

**Figure 8.**Results of fitting to population growth data reflecting metabolic burden caused by IPTG. The plot contains five figures, each showing fitting of the same model to the bacterial population growth data for different concentration of the inducer (IPTG) during 10 h long induction phase. The experimental data are pictured as points with standard error bars, the dashed lines show simulation data for initial values of the model function (i.e., initial point of fitting), the solid lines show the results of fitting and the dotted lines represent the final results optimised by MCMC method of the FME package (Section 2.6), which show the best agreement with the experimental data. The model with the best fit appears to be an enhanced Monod function where the maximum growth rate constant is substituted by the ramp function (defined in Figure 2) going from the maximum growth rate to the minimum growth rate reflecting the metabolic burden effect of the gradually-growing concentration of IPTG. The x-axes show the time of experiment in hours while the y-axes show CDW in g/L.

**Figure 9.**Evidence of exacerbation effect of IPTG on toxicity caused by TCP. Combined effect of metabolic burden caused by 0.2 mM IPTG and toxicity caused by TCP on E. coli BL21(DE3) cells carrying empty plasmids pCDF and pETDuet is displayed as the percentage of survived cells (blue bars) after induction in medium with or without 2 mM TCP. Pre-induced (IPTG +) or non-induced (IPTG −) cells were incubated in buffer with TCP (TCP +) or without (TCP −). The separate negative effects of TCP (orange), IPTG (gray), and the exacerbation of TCP toxicity in cells pre-induced with IPTG (yellow) are indicated. Error bars represent standard deviations calculated from at least five independent experiments. Note that experimental data come from [19].

**Figure 10.**Exacerbation of TCP toxicity in Escherichia coli BL21(DE3) bearing synthetic TCP pathway by IPTG. The leftmost column in each dataset represents the population of cells pre-induced with various concentrations of IPTG. The middle column of each dataset shows the population of survived cells pre-induced with the same amount of IPTG after 5 h in the presence of 2 mM TCP. The rightmost column in each dataset shows the difference of the first and the second column (i.e., $\mathrm{third}=\mathrm{second}-\mathrm{first}$). Note that the population in the first column of the first dataset is pre-induced with 0 mM IPTG and incubated in absence of TCP, which makes it a control group. The second column in the same dataset shows the sole effect of 2 mM TCP on the population. It is remarkable that the third columns in all other datasets seem to be in perfect match and approximately 1.82 times bigger than the same column in the first dataset. We explain this fact by the existence of the exacerbation effect. Different concentrations of IPTG were used for the induction of the TCP pathway expression from pCDF and pETDuet plasmids. Error bars represent standard deviations calculated from at least three independent experiments except for the rigthmost column in each dataset where error bars represent standard error of values in these columns.

**Figure 11.**An ODE model of the extended TCP metabolic pathway. The model represents a chain reaction for biodegradation of TCP into GLY reflecting the E. coli population. Note that the rate constants of the original ODE model were rescaled from seconds (${\mathrm{s}}^{-1}$) into hours (${\mathrm{h}}^{-1}$) because the data used for fitting were sampled every hour. It concerns the original rate constants (${k}_{{1}_{R}}$, ${k}_{{1}_{S}}$, ${k}_{{2}_{R}}$, ${k}_{{2}_{S}}$, ${k}_{3}$, ${k}_{4}$, ${k}_{5}$). Units: ${k}_{\ast},{V}_{\ast},{\mu}_{\ast},{\gamma}_{\ast}\left({\mathrm{h}}^{-1}\right)$; ${t}_{\ast},{K}_{\ast}\left(\mathrm{mM}\right)$; $e{x}_{\ast},Y,{n}_{\ast}\left(\mathrm{unitless}\right)$.

**Figure 12.**Results of parameter synthesis process for Property 3. Inside the figure, one can see two plots with blue regions. Both plots show a combination of parameters and (or) variables of the model where each point represents the particular evaluation of considered parameters (or variables). Every blue region represents a set of evaluations satisfying the stated property in at least one initial condition of the model. Here, the blue regions make joint projections across all non-displayed dimensions (i.e., parameters and variables). Consequently, the property holds in every combination of initial conditions (respectively, parameters) in the particular ranges.

**Figure 13.**Results of parameter synthesis process for Property 5. Inside the figure, one can see two plots with blue regions. Both plots show a combination of parameters and (or) variables of the model where each point represents the particular evaluation of considered parameters (or variables). Every blue region represents a set of evaluations satisfying the stated property in at least one initial condition of the model. Here, the blue regions make joint projections across all non-displayed dimensions (i.e., parameters and variables). Consequently, the property holds in every combination of initial conditions (respectively, parameters) in the particular ranges.

**Figure 14.**Diagram for satisfiability of Property 6 reflecting ${\mathrm{IPTG}}_{0}$. (

**a**) A simple diagram presenting the qualitative results of robustness monitoring for Property 6. It shows the influence of ${\mathrm{IPTG}}_{0}$ on the satisfiability of the particular property. The two thresholds divide the area of influence into three sections. The left one which robustly violates the property, the right one—satisfying the property (robustly)—and the middle one where the result is not robust. This result holds for all combinations of the parameters (and variables) in the table (

**b**).

**Figure 15.**Results of robustness monitoring for Property 6 concerning ${\mathrm{TCP}}_{0}$. The figure shows a two-dimensional plot with various coloured circles pointing by their centre to the particular setting of the plotted parameters (or variables). Initial values of variables and considered parameters (if not displayed in any axis) are: ${\mathrm{Bact}}_{0}=0.487$ (g/L); ${\mathrm{GLY}}_{0}$, $(R{)-\mathrm{DCP}}_{0}$, $(S{)-\mathrm{DCP}}_{0}$, ${\mathrm{ECH}}_{0}$, ${\mathrm{CPD}}_{0}$, ${\mathrm{GDL}}_{0}$, ${\mathrm{TCP}}_{0}=0$ (mM); ${\gamma}_{\mathrm{Bact}}=0.0022$ $\left({\mathrm{h}}^{-1}\right)$. All the constants can be found in Figure 11. The shades of green colour imply a feasibility of the particular property in the particular initial setting while the shades of red imply a violation of the property—the darker the tone, the stronger the feasibility/violation. At the bottom of the plot, there is the feasibility scale mapped to real values. The plot represents a single layer of the entire parameter space.

**Table 1.**Results of robustness monitoring for Property 7. A simple table presents the relevant ranges of initial conditions (i.e., the concentration of variables and setting of parameters) which robustly violate Property 7.

${\mathbf{IPTG}}_{0}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{mM}\right)$ | ${\mathbf{TCP}}_{0}\left(\mathbf{mM}\right)$ | ${\mathbf{Bact}}_{0}(\mathit{g}/\mathbf{L})$ | ${\mathit{\gamma}}_{\mathbf{Bact}}\phantom{\rule{4pt}{0ex}}\left({\mathit{h}}^{-1}\right)$ |
---|---|---|---|

(0.0, 1.0) | (0.0, 4.0) | (0.024, 0.78) | (0.0, 0.01) |

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Demko, M.; Chrást, L.; Dvořák, P.; Damborský, J.; Šafránek, D.
Computational Modelling of Metabolic Burden and Substrate Toxicity in *Escherichia coli* Carrying a Synthetic Metabolic Pathway. *Microorganisms* **2019**, *7*, 553.
https://doi.org/10.3390/microorganisms7110553

**AMA Style**

Demko M, Chrást L, Dvořák P, Damborský J, Šafránek D.
Computational Modelling of Metabolic Burden and Substrate Toxicity in *Escherichia coli* Carrying a Synthetic Metabolic Pathway. *Microorganisms*. 2019; 7(11):553.
https://doi.org/10.3390/microorganisms7110553

**Chicago/Turabian Style**

Demko, Martin, Lukáš Chrást, Pavel Dvořák, Jiří Damborský, and David Šafránek.
2019. "Computational Modelling of Metabolic Burden and Substrate Toxicity in *Escherichia coli* Carrying a Synthetic Metabolic Pathway" *Microorganisms* 7, no. 11: 553.
https://doi.org/10.3390/microorganisms7110553