# Component Characterization in a Growth-Dependent Physiological Context: Optimal Experimental Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Derivation of the Physiological Gene Expression Model

#### 2.1.1. Cell Volume and Mass, DNA Content, and Protein Mass

#### 2.1.2. Total RNA Polymerase (RNAP)

#### 2.1.3. Available RNAP

#### 2.1.4. Transcription Rate

^{−1}[50]. Assuming ${P}_{R}$ exemplifies a strong promoter, we estimate a reasonable range based on analysis of constitutive promoters and mutants of the ${P}_{R}$ sequences as $1-30$ min

^{−1}[50,51,52,53].

#### 2.1.5. mRNA Degradation

^{−3}[63] and taking a nominal half-life of 3 min, we suggest $\delta \approx 2.57\times {10}^{-4}$ μm

^{−3}min

^{−1}.

#### 2.1.6. Total and Free Ribosome Populations

#### 2.1.7. Translation Rate

^{−1}to 10 min

^{−1}.

^{−3}and 3000 μm

^{−3}. We let ${K}_{M}$ range between 750 μm

^{−3}and 1500 μm

^{−3}, with a nominal value of ${K}_{M}=750$ μm

^{−3}, which is approaching saturation (and hence the near constant translation efficiency observed by Klumpp and Bremer [16,17]).

#### 2.2. Optimal Experimental Design

## 3. Results

#### 3.1. Comparing Lumped and Physiologically Aware Models

#### 3.2. Null and Optimal Experimental Designs

#### 3.3. Utility of Optimal Designs for Parameter Identification and Prediction

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Growth media nutrient quality dictates the growth rate of an E. coli culture. The observed growth rate can be used to predict many physiological parameters of the host, which in turn influence gene expression. (

**B**) By accounting for physiological parameters, we can optimally estimate intrinsic parameters of a genetic construct, which reflect properties of its sequence. These intrinsic parameters can be reused across growth conditions and can be used to guide changes to the construct sequence.

**Figure 2.**(

**A**,

**B**) Steady state protein (A) and mRNA (B) copy number predictions for the lumped and full model across growth rate for two different constant input levels. (

**C**,

**D**) Corresponding protein (C) and mRNA (D) concentrations. (

**E**) Relative difference in the lumped parameter values, fit at 3 db/h, compared with the values predicted by the full growth dependent model. (

**F**) Root mean squared protein concentration error over the dynamic simulation (G) between the full and lumped model. (

**G**) Predicted protein concentrations of the lumped and full model over a dynamic simulation with a growth rate of 0.5 db/h.

**Figure 3.**An optimal experiment (designed at the true value of $\mathit{\theta}$). Each column depicts one sub-experiment, labelled with optimal doubling rates: ${\mu}^{\left(1\right)}$, ${\mu}^{\left(2\right)}$ and ${\mu}^{\left(3\right)}$ (corresponding to exponential growth rates ${\lambda}^{\left(1\right)}$, ${\lambda}^{\left(2\right)}$ and ${\lambda}^{\left(3\right)}$). The top row (

**A**–

**C**) depicts the optimal inputs, ${u}^{\left(1\right)}\left(t\right)$, ${u}^{\left(2\right)}\left(t\right)$ and ${u}^{\left(3\right)}\left(t\right)$ (on a ${\mathrm{log}}_{10}$ scale). The middle row (

**D**–

**F**) shows the system response (both mRNA and protein copy number) for each induction profile. The last row shows the sampling densities ${w}_{rna}^{\left(i\right)}$ and ${w}_{prot}^{\left(i\right)}$ (in shaded areas, blue and red, respectively), for both protein and mRNA, as well as the rounded sampling schedule, ${s}_{rna}^{\left(i\right)}$ and ${s}_{prot}^{\left(i\right)}$ (depicted by the stem plot). The optimality score of this experiment was $-69.6$.

**Figure 4.**(

**A**) Results from the true optimal (optimal experiment designed at true parameter value) and null experiments (circles): optimality score on the horizontal axis; observed generalized variance of the parameter estimate on the horizontal axis. (

**B**) The same experiments: generalized parameter variance on the horizontal axis; log integral of the squared error on an out-of-sample prediction experiment (inset) on the vertical axis. Optimal experiments designed with erroneous initial parameter guesses are shown as X’s. (Linear trend lines also shown).

Parameter Label | Intrinsic Parameter | Nominal Value | Feasible Range |
---|---|---|---|

Intrinsic Transcription Parameters | |||

Promoter Escape Rate | $\alpha $ | 20 min^{−1} | [1–30] |

RNAP-Promoter Binding | ${K}_{r}$ | 40 | [10–40] |

TF-Promoter Binding | ${K}_{t}$ | $5\times {10}^{5}$ | [$2\times {10}^{3}$–$1\times {10}^{6}$] |

TF-RNAP Interaction | ${K}_{rt}$ | $1.09\times {10}^{9}$ | [$4.02\times {10}^{5}$–$5.93\times {10}^{10}$] |

Intrinsic mRNA Decay Parameters | |||

mRNA Decay Rate | $\delta $ | $2.57\times {10}^{-4}$μm^{−3} min^{−1} | [$7.7\times {10}^{-5}$–$7.7\times {10}^{-4}$] |

Intrinsic Translation Parameters | |||

Max. Initiation Rate | $\beta $ | $4.0$ min^{−1} | [1–10] |

Half-saturating Constant | ${K}_{M}$ | 750 μm^{−3} | [750–1500] |

Property Label | Physiological Property | Value at μ = 0.6 db/h | Value at μ = 3 db/h |

Physiological Properties of Transcription | |||

Gene Copy Number | g | 1.4 | 5.7 |

Available RNAP | ${P}_{a}$ | 1000 | 4000 |

Genome-lengths of DNA | G | 1.3 | 4.3 |

Physiological Properties of mRNA Decay | |||

RNase Concentration | $\xi $ | 900 μm^{−3} min^{−1} | 900 μm^{−3} min^{−1} |

Physiological Properties of Translation | |||

Free Ribosomes | ${R}_{f}$ | 600 | 7000 |

General Physiological Properties | |||

Cell Volume | V | 0.4 μm^{−3} | 2.24 μm^{−3} |

Growth Rate | $\lambda $ | $0.7\times {10}^{-2}$ min^{−1} | $3.5\times {10}^{-2}$ min^{−1} |

Experiment | Growth Rates (db/h) | Induction Pattern (log_{10}) | Sampling Schedule | Optimality |
---|---|---|---|---|

Null Experiment | {0.6,1.8,3} | −63.8 | ||

Growth Variant | {2,2.5,3} | −61.5 | ||

Sampling Variant | {0.6,1.8,3} | −62.6 | ||

Induction Variant | {0.6,1.8,3} | −60.5 |

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**MDPI and ACS Style**

Braniff, N.; Scott, M.; Ingalls, B.
Component Characterization in a Growth-Dependent Physiological Context: Optimal Experimental Design. *Processes* **2019**, *7*, 52.
https://doi.org/10.3390/pr7010052

**AMA Style**

Braniff N, Scott M, Ingalls B.
Component Characterization in a Growth-Dependent Physiological Context: Optimal Experimental Design. *Processes*. 2019; 7(1):52.
https://doi.org/10.3390/pr7010052

**Chicago/Turabian Style**

Braniff, Nathan, Matthew Scott, and Brian Ingalls.
2019. "Component Characterization in a Growth-Dependent Physiological Context: Optimal Experimental Design" *Processes* 7, no. 1: 52.
https://doi.org/10.3390/pr7010052