# Mechanistic Models of Inducible Synthetic Circuits for Joint Description of DNA Copy Number, Regulatory Protein Level, and Cell Load

^{1}

^{2}

^{*}

## Abstract

**:**

_{lux}) concentrations, inducer-protein complex formation, and resource usage limitation. We demonstrated that a change in the considered model assumptions can significantly affect circuit output, and preliminary experimental data are in accordance with the simulated activation curves. We finally showed that the models are identifiable a priori (in the analytically tractable cases) and a posteriori, and we determined the specific experiments needed to parametrize them. Although a larger-scale experimental validation is required, in the future the reported models may support synthetic circuits output prediction in practical situations with unprecedented details.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Inducible System Description

_{lux}promoter in presence of N-(3- oxohexanoyl)-L-homoserine lactone (HSL). The LuxR-HSL active complex binds to the lux box of the P

_{lux}promoter, triggering its activity in an HSL concentration-dependent fashion, thereby regulating the transcription of the downstream gene, which is then translated into protein. We considered a set of reactions affecting the production of the (reporter) protein-encoded gene regulated by P

_{lux}, similar to the ones described by Carbonell-Ballestero et al. [20] (Figure 1b). Specifically, a LuxR protein dimer (${R}_{2}$) and HSL ($L$) form a complex ($Q$), which binds the free promoter ($P$) to form a transcriptionally active bound promoter ($S$).

#### 2.2. Model Definition

#### 2.2.1. Empirical Michaelis–Menten Model (M0)

#### Assumptions

_{lux}and the mature form of the protein ($Y$) can be described via Equations (1)–(3):

^{−1}) units. In Equation (2), ${\gamma}_{X}$ (time

^{−1}) includes the protein degradation and dilution rates, and $\rho $ (time

^{−1}) is the translation rate. If the expressed protein is very stable, ${\gamma}_{X}$ is equal to the rate of cell division. When relevant in terms of dynamics, protein maturation or folding is also included: $Y$ represents mature protein, $\sigma $ (time

^{−1}) represents protein maturation rate and protein degradation rate is assumed to be the same for immature and mature forms. Assuming the steady-state of all the intracellular processes, the output commonly considered for such system, i.e., the mature protein synthesis rate per cell ($y$), which is equal to the synthesis term of Equation (3), is proportional to $H\left(L\right)$ (Equation (4))

_{lux}is in its off state;$\alpha $ is the activity range in the on state;$\kappa $ is the HSL concentration corresponding to half of the maximum activation [20]. The $\delta $ and $\alpha $ parameters are expressed as intracellular protein concentration per time. The experimental measurements routinely performed in laboratory to characterize synthetic circuits usually exploit fluorescent reporter proteins, which are quantified via in vivo assays by means of plate reader or flow-cytometry. In these cases, per-cell arbitrary units of fluorescence (AU) can be adopted to express intracellular protein concentration, assuming their proportionality.

#### Derivation

#### Final Expression

_{lux}(${n}_{1}$) and LuxR (${n}_{2}$) can be included in the model. Specifically, ${n}_{1}$ and ${n}_{2}$ can act as known scale factors of $\widehat{\delta}$, $\widehat{\alpha}$ and $U$ (Equations (12)–(14)), thereby enabling the description of copy number changes among different situations. Following a bottom-up approach, model parameters ($\widehat{\delta}$ or $d$, $\widehat{\alpha}$ or $v$, $\widehat{\kappa}$ and ${\kappa}_{R}$) can be estimated from experimental data and the parametrized model can be adopted to predict unseen situations with different ${n}_{1}$ and ${n}_{2}$

^{-1}, like $\widehat{\delta}$ and $\widehat{\alpha}$. Analogously, $U$ and $\widehat{\kappa}$ are intracellular protein concentrations that can also be expressed in AU, while ${\kappa}_{R}$ has the same units as HSL concentration. For this reason, the relative copy number changes, instead of the absolute ones, are sufficient to express ${n}_{1}$ and ${n}_{2}$. Specifically, ${n}_{1}$ can be set according to plasmid copy number and ${n}_{2}$ is proportional to the strength of the promoter expressing LuxR. Finally, the $u$ value (Equation (14)) can be set to 1 without any loss of generality, since $U$ is always present in ratio with $\widehat{\kappa}$ (Equation (8)). The final empirical model, called M0, describing the output of the lux inducible system as a function of HSL and the per-cell copy numbers of P

_{lux}and LuxR, is reported in Equation (15)

#### 2.2.2. Mechanistic Model with LuxR Abundance Assumption (M1)

#### Assumptions

_{lux}, and the low-entity promoter activation by this unspecific complex is neglected. Previous experimental work showed that this activity increase is undetectable in the commonly tested conditions [35].

#### Derivation

#### Final Expression

_{lux}copy number, and the scale factor between one protein monomer and the actual dimer concentration.

#### 2.2.3. Mechanistic Model without LuxR Abundance Assumption (M2)

#### Assumptions and Final Expression

#### 2.2.4. Modeling LuxR-HSL Hetero-Tetramerization (M1T, M2T)

#### Assumptions

#### Derivation

#### Final Expression

#### 2.2.5. Modeling Cell Load (M1L, M2L)

#### Assumptions

_{lux}is characterized by a relevant resource usage, while LuxR does not cause relevant burden. Since LuxR is constitutively expressed, thereby giving constant load, this assumption will not affect any HSL-dependent function, as previously discussed for genes with constant resource usage acting as background [23].

#### Derivation

#### Final Expression

#### 2.3. Model Analysis

#### 2.3.1. Model Parametrization

_{lux}activation curve, resource usage [23,31,37]). A summary of parameters is provided in Table 1 for each model used in this work. When indicated, structural parameters were fixed to the reported values in simulations and they were assumed to be unknown during parameter estimation tasks (e.g., when studying a posteriori identifiability). E. coli cell volume was assumed to be 1 µm

^{3}, corresponding to 10

^{−15}L. Under this assumption, the concentration of one molecule of promoter DNA or protein corresponds to 1.66 nM [38]. A variation of the average E. coli cell volume has been previously reported in the range 0.5‒2 µm

^{3}for different growth rates and conditions [38]. This variation is expected to affect the absolute values but not the trends of the numerical simulations shown of this work, thereby not affecting the drawn conclusions. Nonetheless, the variation of cell volume will be highly relevant in the analysis of real data, in which model parameters have to be estimated based on the knowledge of this value.

^{−1}), the M1 model can be re-parametrized as Equation (45)

^{−1}), thereby yielding the model in Equation (46)

#### 2.3.2. A Priori Identifiability

#### 2.3.3. A Posteriori Identifiability

#### 2.3.4. Simulations

#### 2.3.5. Analysis of Activation Curves

#### 2.3.6. In Vivo Experiments

_{cell}) at steady-state, was measured for recombinant MG1655-Z1 strain [31] bearing the low-copy plasmid pSB4C5 with X

_{3}r as insert [23]. In this construct, previously described in [23] and with sequence available as BBa_J107032 code in the Registry of Standard Biological Parts (http://parts.igem.org), luxR is under the control of an anhydrotetracycline (ATc) inducible promoter, which works as gene expression knob in Escherichia coli strains overexpressing TetR, like the one used in this study.

_{cell}measurement was previously described [23]. Briefly, 0.5 ml of M9 medium (11.28 g/L M9 salts, 1 mM thiamine hydrochloride, 2 mM MgSO

_{4}, 0.1 mM CaCl

_{2}, 0.2% casamino acids and 0.4% glycerol) were inoculated with a colony from a freshly streaked LB agar plate in a 2-ml tube and the culture was incubated at 37 °C, 220 rpm for at least 16 h. The grown culture was 100-fold diluted in 200 µL of M9 in a 96-well microplate. ATc and HSL (2 µl) were added to the microplate wells to reach the desired concentrations. The microplate was incubated at 37 °C in the Infinite F200 (Tecan) microplate reader and an automatic measurement procedure was programmed via the i-control software v.2.0.10 (Tecan, Switzerland): shaking (15 s, 3-mm amplitude), wait (5 s), optical density (600 nm) acquisition, red fluorescence (excitation: 535 nm, emission: 620 nm, gain = 50) acquisition, sampling time = 5 min. Raw data time series were background-subtracted using sterile medium (absorbance) and a non-fluorescent culture (fluorescence), incubated in the same experiment. The resulting data were used to compute S

_{cell}as the numeric time derivative of fluorescence, divided by absorbance over time. S

_{cell}was averaged in the exponential growth phase, typically occurring at absorbance values between 0.05 and 0.18 [32]. Finally, the average S

_{cell}values of the X

_{3}r strains at the desired inductions were normalized by the S

_{cell}value of a reference culture (MG1655-Z1 bearing the BBa_J107029 constitutive RFP expression cassette in the pSB4C5 plasmid) as internal control to obtain S

_{cell}in standardized relative units.

## 3. Results

#### 3.1. Simulated Output for Different Model Assumptions

#### 3.1.1. Effect of LuxR Abundance Assumption: M1 vs. M2

_{lux}(Figure 2).

_{2T}decreases below the promoter concentration value (about 8 nM) the output of M2 showed both lower $\alpha $ and $\kappa $ levels compared to M1, with the decrease of $\kappa $ showing the highest-entity effect (>2-fold with the used parameters, Figure 2e). When plasmid copy number is varied in a range below a constant LuxR concentration value (500 nM), $\alpha $ still showed a linear increase as observed in M1, but, differently from M1, $\kappa $ showed a low-entity increase (less than 2-fold). For concentrations of promoter higher than LuxR, the $\alpha $ value showed saturation, intuitively because all the P

_{lux}promoters in the cell cannot be occupied by the limiting amount of LuxR, and as a result RFP synthesis cannot increase anymore for higher concentrations of promoter. Although in this latter condition RFP maximum expression is constant, an increase in plasmid copy number results in a decrease of $\kappa $ (about 2-fold). Also, in the M2 model, the Hill coefficient was equal to 1 upon ${R}_{2T}$ variation. However, interestingly, it decreased to values slightly lower than 1 (up to 0.8) upon variations of plasmid copy number (Figure 2g,h).

#### 3.1.2. Effects of Cell Load

#### 3.1.3. Evaluation of LuxR-HSL Complex Formation Assumptions: M(1-2) vs. M(1-2)T

#### 3.2. Model Identifiability

#### 3.2.1. M1

#### 3.2.2. M2

_{2}= 0.005 (corresponding to ${R}_{2T}$ = 0.5 nM) and, as before, 0.5 and 5 AU, to estimate ${K}_{3}$.and ${r}_{T}$, exploiting at least one condition in which LuxR is not abundant compared with the P

_{lux}promoter. This procedure led to a structurally identifiable condition for the M2 model, since it could estimate its parameters with reasonably low estimation error and CV, even if it had higher REE compared with M1 but much lower CV (Figure 5).

#### 3.2.3. M1T

#### 3.2.4. M2T

## 4. Discussion

_{lux}, with a resulting mathematical expression equivalent to a Michaelis–Menten function model.

_{lux}copy number was predicted to result into a linearly increasing expression by M1, while M2 showed a saturating trend, thereby demonstrating that the removal of this simplifying assumption can lead to an intuitively more realistic behavior of the inducible system upon DNA copy number changes. In this situation and with the structural parameter values used in our study, the output showed maximum activity variations up to 2.3-fold between M1 and M2 for biologically plausible values of DNA and protein concentrations.

_{lux}concentration had to be measured. Considering a proportional error model for the generated data (5% CV), all the models enabled the estimation of structural parameters within 2-fold compared with the true value (considering interquartile ranges of REE distribution). The M2 model showed the highest estimation error (>100%, median among 200 runs with different datasets simulated with a 10% proportional error), making it the less robust model among the tested ones in estimation tasks. The M1T model with a single LuxR level showed similar drawbacks, which could be overcome by adding activation curve data with more LuxR levels, while M2 could not be improved following the same procedure.

_{lux}module, considered in this work. The underlying binding reactions could be known, or they might be investigated by testing different model assumptions against experimental data for model parametrization and mechanistic understanding.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Description of the lux inducible system. (

**a**) The gene encoding the LuxR protein transcriptional regulator is expressed by a constitutive promoter (P

_{con}); the LuxR regulator becomes activated upon HSL molecule binding to form a complex which can bind the P

_{lux}promoter in its single lux box sequence, thereby activating the expression of the gene of interest (GOI), placed downstream of P

_{lux}. Curved arrows represent promoters; ovals represent ribosome binding sites (RBSs); straight arrows represent coding sequences; hexagons represent transcriptional terminators; circles represent HSL molecules. (

**b**,

**c**) Biomolecular reactions modeled in this work. The LuxR dimer (R

_{2T}, blue overlapping circles) is assumed to be activated upon binding of one (

**b**) or two (

**c**) HSL molecules (L, yellow circles) to form an activated complex (Q or Q

_{2}, respectively), which binds DNA (double helix icon) to enable transcription. In panel (

**c**), Q is the LuxR dimer form bound to one HSL molecule and it is assumed to be unable to bind DNA. The equilibrium constants are reported for each reaction occurring in panels (

**b**,

**c**).

**Figure 2.**Comparison between M1 and M2. Panels (

**a**,

**b**) report the output activation curves of M1 (solid lines) and M2 (dashed lines) for different values of R

_{2T}(panel (

**a**), as indicated in the legend, expressed as nM) and P

_{T}(panel (

**b**), as indicated in the legend, expressed as per cell copy number). The (

**c**)–(

**d**), (

**e**)–(

**f**), and (

**g**)–(

**h**) panel pairs report the R

_{2T}- and P

_{T}-dependent trend of α, κ and η, respectively, with solid and dashed lines representing M1 and M2, respectively. In panels (

**a**,

**c**,

**e**,

**g**) the promoter copy number value was set to 5, while in panels (

**b**,

**d**,

**f**,

**h**) the LuxR concentration was set to 500 nM.

**Figure 3.**Comparison between M1 and M1L. Panels (

**a**,

**b**) report the output activation curves of M1 (solid lines) and M1L (dashed lines) and M1L with external load, indicated as M1L+E (dotted lines), for different values of R

_{2T}(panel (

**a**), as indicated in the legend, expressed as nM) and P

_{T}(panel (

**b**), as indicated in the legend, expressed as per cell copy number). The (

**c**)–(

**d**), (

**e**)–(

**f**), and (

**g**)–(

**h**) panel pairs report the R

_{2T}- and P

_{T}-dependent trend of α, κ and η, respectively, with solid, dashed, and dotted lines representing M1, M1L, and M1L+E, respectively. In panels (

**a**,

**c**,

**e**,

**g**) the promoter copy number value was set to 5, while in panels (

**b**,

**d**,

**f**,

**h**) the LuxR concentration was set to 500 nM.

**Figure 4.**Comparison between M1T and M2T. Panels (

**a**,

**b**) report the output activation curves of M1T (solid lines) and M2T (dashed lines) for different values of R

_{2T}(panel (

**a**), as indicated in the legend, expressed as nM) and P

_{T}(panel (

**b**), as indicated in the legend, expressed as per cell copy number). The (

**c**)–(

**d**), (

**e**)–(

**f**), and (

**g**)–(

**h**) panel pairs report the R

_{2T}- and P

_{T}-dependent trend of α, κ and η, respectively, with solid and dashed lines representing M1T and M2T, respectively. In panels (

**a**,

**c**,

**e**,

**g**) the promoter copy number value was set to 5, while in panels (

**b**,

**d**,

**f**,

**h**) the LuxR concentration was set to 500 nM.

**Figure 5.**Relative estimation error (REE) and uncertainty of parameter estimates (CV) in a posteriori identifiability. Panels (

**a**,

**b**) report the distribution of REE (

**a**) and CV (

**b**) among 200 runs starting from different simulated data, with a 5% proportional error. The model and the number of LuxR levels included in the fitted data are specified for each boxplot. The red line represents the median of the distribution. Panels (

**c**,

**d**) report the median of the REE (

**c**) and CV (

**d**) distribution as a function of the proportional error entity, from 0 to 10%. The models and specific LuxR levels are described in the legend. In all the panels, the dashed horizontal line indicates the 100% value to facilitate the interpretation of the graphs.

**Table 1.**Model parameters. The indicated values were used for simulation and, unless differently indicated, for the study of a posteriori identifiability.

Parameter | Units | Models | Value |
---|---|---|---|

P_{U} | nM | All | 1.66 |

r_{T} | nM ^{a} | All | 1.66 |

n_{1} | - | All | 5 ^{b} |

n_{2} | - ^{a} | All | 301 ^{b} |

γ_{X} | min^{−1} | All | 0.01 ^{c} |

σ | min^{−1} | All | 0.0167 |

${\widehat{k}}_{m0}$ | AU nM^{−1} min^{−1} | All | 0.0192 |

${\widehat{k}}_{mL}$ | AU nM^{−1} min^{−1} | All | 1.907 |

K_{1} | nM^{−1} | M1, M1L, M2, M2L | 0.001 |

M1T, M1TL, M2T, M2TL | 0.0055 | ||

K_{3} | nM^{−1} | M1, M1L, M2, M2L | 0.2 |

K_{4} | nM^{−1} | M1T, M2T, M1TL, M2TL | 0.001375 |

K_{5} | nM^{−1} | M1T, M2T, M1TL, M2TL | 0.2 |

J_{RFP} | min AU^{−1} | M1L, M2L, M1TL, M2TL | 0.04 |

E | - | M1L, M2L, M1TL, M2TL | 0 ^{d} |

^{a}During the study of a posteriori identifiability, n

_{2}was expressed as AU with known value, while r

_{T}was expressed as nM/AU with unknown value;

^{b}When indicated, a range of copy number values was spanned;

^{c}Typical value of E. coli growth rate, which is used as the rate of intracellular protein dilution;

^{d}A value of 2 was used to simulate the presence of external load where indicated.

**Table 2.**Parameters estimated from in vivo experiments. Measurements were carried out using MG1655-Z1 as host strain.

Condition | α (%) ^{a} | κ (nM) | η (-) | Reference |
---|---|---|---|---|

Medium copy ^{b}, no IPTG | 47 | 194.01 | 1.01 | [31] |

Medium copy + E ^{c}, no IPTG | 27 | 474.1 | 0.98 | [31] |

Medium copy, IPTG = 500 µM | 100 | 0.77 | 1.54 | [31] |

Medium copy + E, IPTG = 500 µM | 68 | 0.99 | 1.29 | [31] |

Low copy ^{d}, no ATc | 83 | 34.29 | 0.98 | This study |

Low copy, ATc = 2.5 ng/ml | 92 | 11.32 | 1 | This study |

Low copy, ATc = 5 ng/ml | 97 | 3.73 | 1.15 | This study |

Low copy, ATc = 50 ng/ml | 100 | 1.52 | 1.39 | This study |

^{a}Percent expression, relative to the maximum expression value obtained in the same study;

^{b}BioBrick™ construct BBa_J107063 in the pSB3K3 medium-copy vector, IPTG-inducible LuxR expression cassette;

^{c}Additional load provided by the OL1 low-copy plasmid, described in the same paper;

^{d}BioBrick™ construct BBa_J107032 in the pSB4C5 low-copy vector, ATc-inducible LuxR expression cassette.

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**MDPI and ACS Style**

Pasotti, L.; Bellato, M.; De Marchi, D.; Magni, P.
Mechanistic Models of Inducible Synthetic Circuits for Joint Description of DNA Copy Number, Regulatory Protein Level, and Cell Load. *Processes* **2019**, *7*, 119.
https://doi.org/10.3390/pr7030119

**AMA Style**

Pasotti L, Bellato M, De Marchi D, Magni P.
Mechanistic Models of Inducible Synthetic Circuits for Joint Description of DNA Copy Number, Regulatory Protein Level, and Cell Load. *Processes*. 2019; 7(3):119.
https://doi.org/10.3390/pr7030119

**Chicago/Turabian Style**

Pasotti, Lorenzo, Massimo Bellato, Davide De Marchi, and Paolo Magni.
2019. "Mechanistic Models of Inducible Synthetic Circuits for Joint Description of DNA Copy Number, Regulatory Protein Level, and Cell Load" *Processes* 7, no. 3: 119.
https://doi.org/10.3390/pr7030119